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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ot3rdgg 5801 | Extract the third member of an ordered triple. (See ot1stg 5799 comment.) (Contributed by NM, 3-Apr-2015.) |
Theorem | 1stval2 5802 | 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 5803 | 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 5804 | The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | fo2nd 5805 | The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | f1stres 5806 | 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 5807 | 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 5808* | Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Theorem | fo2ndresm 5809* | Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
Theorem | 1stcof 5810 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
Theorem | 2ndcof 5811 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
Theorem | xp1st 5812 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | xp2nd 5813 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | 1stexg 5814 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Theorem | 2ndexg 5815 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Theorem | elxp6 5816 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4828. (Contributed by NM, 9-Oct-2004.) |
Theorem | elxp7 5817 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4828. (Contributed by NM, 19-Aug-2006.) |
Theorem | eqopi 5818 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Theorem | xp2 5819* | Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
Theorem | unielxp 5820 | The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
Theorem | 1st2nd2 5821 | Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
Theorem | xpopth 5822 | An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.) |
Theorem | eqop 5823 | 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 5824 | Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
Theorem | op1steq 5825* | 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 5826 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
Theorem | 1st2nd 5827 | Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
Theorem | 1stdm 5828 | 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 5829 | 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 5830 | Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Theorem | releldm2 5831* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Theorem | reldm 5832* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Theorem | sbcopeq1a 5833 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2824 that avoids the existential quantifiers of copsexg 3999). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | csbopeq1a 5834 | Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 2916). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfopab2 5835* | 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 5836* | 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 5837* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
Theorem | dfoprab4 5838* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | dfoprab4f 5839* | 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 5840* | Define the cross product of three classes. Compare df-xp 4369. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
Theorem | elopabi 5841* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
Theorem | eloprabi 5842* | 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 5843* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | mpt2mpts 5844* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.) |
Theorem | dmmpt2ssx 5845* | The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Theorem | fmpt2x 5846* | 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 5847* | Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | fnmpt2 5848* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | mpt2fvex 5849* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | fnmpt2i 5850* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | dmmpt2 5851* | Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
Theorem | mpt2fvexi 5852* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | mpt2exxg 5853* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
Theorem | mpt2exg 5854* | Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Theorem | mpt2exga 5855* | 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 5856* | 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 5857* | Composition of two functions. Variation of fmptco 5351 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Theorem | oprabco 5858* | 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 5859* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
Theorem | df1st2 5860* | An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | df2nd2 5861* | An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | 1stconst 5862 | The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.) |
Theorem | 2ndconst 5863 | The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.) |
Theorem | dfmpt2 5864* | Alternate definition for the "maps to" notation df-mpt2 5537 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Theorem | cnvf1olem 5865 | Lemma for cnvf1o 5866. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Theorem | cnvf1o 5866* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Theorem | f2ndf 5867 | 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 5868 | 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 5869 | 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 5870 | Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Theorem | algrflemg 5871 | Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | xporderlem 5872* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
Theorem | poxp 5873* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
Theorem | spc2ed 5874* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
Theorem | cnvoprab 5875* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
Theorem | f1od2 5876* | 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 5537) 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 5602, ovmpt2x 5649 and fmpt2x 5846). 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 5877* | 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 5878* | 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 5879* | 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 5880* | 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 5881* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
Syntax | ctpos 5882 | The transposition of a function. |
tpos | ||
Definition | df-tpos 5883* | Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposss 5884 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeq 5885 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeqd 5886 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
tpos tpos | ||
Theorem | tposssxp 5887 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
tpos | ||
Theorem | reltpos 5888 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos2 5889 | Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos0 5890 | 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 5891 | Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtposg 5892 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
tpos | ||
Theorem | ottposg 5893 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
tpos | ||
Theorem | dmtpos 5894 | The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | rntpos 5895 | The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposexg 5896 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | ovtposg 5897 | The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from ( 1 ... m ) ( 1 ... n ) to the reals or some ring, tpos is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposfun 5898 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos2 5899* | Alternate definition of tpos when has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos3 5900* | Alternate definition of tpos when has relational domain. Compare df-cnv 4371. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos |
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