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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | op1std 5901 | Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Theorem | op2ndd 5902 | Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Theorem | op1stg 5903 | Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Theorem | op2ndg 5904 | Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Theorem | ot1stg 5905 | 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 5905, ot2ndg 5906, ot3rdgg 5907.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
Theorem | ot2ndg 5906 | Extract the second member of an ordered triple. (See ot1stg 5905 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
Theorem | ot3rdgg 5907 | Extract the third member of an ordered triple. (See ot1stg 5905 comment.) (Contributed by NM, 3-Apr-2015.) |
Theorem | 1stval2 5908 | 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 5909 | 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 5910 | The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | fo2nd 5911 | The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | f1stres 5912 | 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 5913 | 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 5914* | Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Theorem | fo2ndresm 5915* | Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Theorem | 1stcof 5916 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
Theorem | 2ndcof 5917 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
Theorem | xp1st 5918 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | xp2nd 5919 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | 1stexg 5920 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Theorem | 2ndexg 5921 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Theorem | elxp6 5922 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4905. (Contributed by NM, 9-Oct-2004.) |
Theorem | elxp7 5923 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4905. (Contributed by NM, 19-Aug-2006.) |
Theorem | eqopi 5924 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Theorem | xp2 5925* | Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
Theorem | unielxp 5926 | The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
Theorem | 1st2nd2 5927 | Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
Theorem | xpopth 5928 | An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.) |
Theorem | eqop 5929 | 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 5930 | Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
Theorem | op1steq 5931* | 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 5932 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
Theorem | 1st2nd 5933 | Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
Theorem | 1stdm 5934 | 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 5935 | 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 5936 | Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Theorem | releldm2 5937* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Theorem | reldm 5938* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Theorem | sbcopeq1a 5939 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2847 that avoids the existential quantifiers of copsexg 4062). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | csbopeq1a 5940 | Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 2939). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfopab2 5941* | 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 5942* | 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 5943* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
Theorem | dfoprab4 5944* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfoprab4f 5945* | 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 5946* | Define the cross product of three classes. Compare df-xp 4434. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
Theorem | elopabi 5947* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
Theorem | eloprabi 5948* | 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 5949* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | mpt2mpts 5950* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.) |
Theorem | dmmpt2ssx 5951* | The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Theorem | fmpt2x 5952* | 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 5953* | Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Theorem | fnmpt2 5954* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Theorem | mpt2fvex 5955* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | fnmpt2i 5956* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Theorem | dmmpt2 5957* | Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Theorem | mpt2fvexi 5958* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | mpt2exxg 5959* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
Theorem | mpt2exg 5960* | Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Theorem | mpt2exga 5961* | 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 5962* | 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 5963* | Composition of two functions. Variation of fmptco 5448 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Theorem | oprabco 5964* | 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 5965* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
Theorem | df1st2 5966* | An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | df2nd2 5967* | An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | 1stconst 5968 | The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.) |
Theorem | 2ndconst 5969 | The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.) |
Theorem | dfmpt2 5970* | Alternate definition for the maps-to notation df-mpt2 5639 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | cnvf1olem 5971 | Lemma for cnvf1o 5972. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Theorem | cnvf1o 5972* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Theorem | f2ndf 5973 | 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 5974 | 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 5975 | 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 5976 | Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Theorem | algrflemg 5977 | Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | xporderlem 5978* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
Theorem | poxp 5979* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
Theorem | spc2ed 5980* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
Theorem | cnvoprab 5981* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
Theorem | f1od2 5982* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
Theorem | disjxp1 5983* | The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Disj Disj | ||
Theorem | disjsnxp 5984* | The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Disj | ||
The following theorems are about maps-to operations (see df-mpt2 5639) 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 5708, ovmpt2x 5755 and fmpt2x 5952). 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 | opeliunxp2f 5985* | Membership in a union of Cartesian products, using bound-variable hypothesis for instead of distinct variable conditions as in opeliunxp2 4564. (Contributed by AV, 25-Oct-2020.) |
Theorem | mpt2xopn0yelv 5986* | 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 5987* | 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 5988* | 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 5989* | 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 5990* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
Syntax | ctpos 5991 | The transposition of a function. |
tpos | ||
Definition | df-tpos 5992* | Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposss 5993 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeq 5994 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeqd 5995 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
tpos tpos | ||
Theorem | tposssxp 5996 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
tpos | ||
Theorem | reltpos 5997 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos2 5998 | Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos0 5999 | 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 6000 | Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos |
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