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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | op2nd 5801 | Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
Theorem | op1std 5802 | Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Theorem | op2ndd 5803 | Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Theorem | op1stg 5804 | Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Theorem | op2ndg 5805 | Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Theorem | ot1stg 5806 | Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5806, ot2ndg 5807, ot3rdgg 5808.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
Theorem | ot2ndg 5807 | Extract the second member of an ordered triple. (See ot1stg 5806 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
Theorem | ot3rdgg 5808 | Extract the third member of an ordered triple. (See ot1stg 5806 comment.) (Contributed by NM, 3-Apr-2015.) |
Theorem | 1stval2 5809 | Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.) |
Theorem | 2ndval2 5810 | Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.) |
Theorem | fo1st 5811 | The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | fo2nd 5812 | The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | f1stres 5813 | Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | f2ndres 5814 | Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | fo1stresm 5815* | Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Theorem | fo2ndresm 5816* | Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Theorem | 1stcof 5817 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
Theorem | 2ndcof 5818 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
Theorem | xp1st 5819 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | xp2nd 5820 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | 1stexg 5821 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Theorem | 2ndexg 5822 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Theorem | elxp6 5823 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4835. (Contributed by NM, 9-Oct-2004.) |
Theorem | elxp7 5824 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4835. (Contributed by NM, 19-Aug-2006.) |
Theorem | eqopi 5825 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Theorem | xp2 5826* | Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
Theorem | unielxp 5827 | The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
Theorem | 1st2nd2 5828 | Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
Theorem | xpopth 5829 | An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.) |
Theorem | eqop 5830 | Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
Theorem | eqop2 5831 | Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
Theorem | op1steq 5832* | Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Theorem | 2nd1st 5833 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
Theorem | 1st2nd 5834 | Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
Theorem | 1stdm 5835 | The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.) |
Theorem | 2ndrn 5836 | The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.) |
Theorem | 1st2ndbr 5837 | Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Theorem | releldm2 5838* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Theorem | reldm 5839* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Theorem | sbcopeq1a 5840 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2795 that avoids the existential quantifiers of copsexg 4008). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | csbopeq1a 5841 | Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 2887). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfopab2 5842* | A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfoprab3s 5843* | A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfoprab3 5844* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
Theorem | dfoprab4 5845* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfoprab4f 5846* | Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfxp3 5847* | Define the cross product of three classes. Compare df-xp 4378. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
Theorem | elopabi 5848* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
Theorem | eloprabi 5849* | A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.) |
Theorem | mpt2mptsx 5850* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | mpt2mpts 5851* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.) |
Theorem | dmmpt2ssx 5852* | The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Theorem | fmpt2x 5853* | Functionality, domain and codomain of a class given by the "maps to" notation, where is not constant but depends on . (Contributed by NM, 29-Dec-2014.) |
Theorem | fmpt2 5854* | Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | fnmpt2 5855* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | mpt2fvex 5856* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | fnmpt2i 5857* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | dmmpt2 5858* | Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | mpt2fvexi 5859* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | mpt2exxg 5860* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
Theorem | mpt2exg 5861* | Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Theorem | mpt2exga 5862* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.) |
Theorem | mpt2ex 5863* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.) |
Theorem | fmpt2co 5864* | Composition of two functions. Variation of fmptco 5357 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Theorem | oprabco 5865* | Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
Theorem | oprab2co 5866* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
Theorem | df1st2 5867* | An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | df2nd2 5868* | An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | 1stconst 5869 | The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.) |
Theorem | 2ndconst 5870 | The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.) |
Theorem | dfmpt2 5871* | Alternate definition for the "maps to" notation df-mpt2 5544 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | cnvf1olem 5872 | Lemma for cnvf1o 5873. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Theorem | cnvf1o 5873* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Theorem | f2ndf 5874 | The (second member of an ordered pair) function restricted to a function is a function of into the codomain of . (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Theorem | fo2ndf 5875 | The (second member of an ordered pair) function restricted to a function is a function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Theorem | f1o2ndf1 5876 | The (second member of an ordered pair) function restricted to a one-to-one function is a one-to-one function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Theorem | algrflem 5877 | Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Theorem | algrflemg 5878 | Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | xporderlem 5879* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
Theorem | poxp 5880* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
Theorem | spc2ed 5881* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
Theorem | cnvoprab 5882* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
Theorem | f1od2 5883* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
The following theorems are about maps-to operations (see df-mpt2 5544) where the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 5609, ovmpt2x 5656 and fmpt2x 5853). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short. | ||
Theorem | mpt2xopn0yelv 5884* | If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
Theorem | mpt2xopoveq 5885* | Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
Theorem | mpt2xopovel 5886* | Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
Theorem | sprmpt2 5887* | The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
Theorem | isprmpt2 5888* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
Syntax | ctpos 5889 | The transposition of a function. |
tpos | ||
Definition | df-tpos 5890* | Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposss 5891 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeq 5892 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeqd 5893 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
tpos tpos | ||
Theorem | tposssxp 5894 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
tpos | ||
Theorem | reltpos 5895 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos2 5896 | Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos0 5897 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | reldmtpos 5898 | Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtposg 5899 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
tpos | ||
Theorem | ottposg 5900 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
tpos |
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