P. 62 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ot1stg 6101	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 6101, ot2ndg 6102, ot3rdgg 6103.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A )
Theorem	ot2ndg 6102	Extract the second member of an ordered triple. (See ot1stg 6101 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B )
Theorem	ot3rdgg 6103	Extract the third member of an ordered triple. (See ot1stg 6101 comment.) (Contributed by NM, 3-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B , C >. ) = C )
Theorem	1stval2 6104	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.)
|- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )
Theorem	2ndval2 6105	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.)
|- ( A e. ( _V X. _V ) -> ( 2nd ` A ) = |^| |^| |^| `' { A } )
Theorem	fo1st 6106	The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- 1st : _V -onto-> _V
Theorem	fo2nd 6107	The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- 2nd : _V -onto-> _V
Theorem	f1stres 6108	Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
Theorem	f2ndres 6109	Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
Theorem	fo1stresm 6110*	Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
|- ( E. y y e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )
Theorem	fo2ndresm 6111*	Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
|- ( E. x x e. A -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )
Theorem	1stcof 6112	Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
|- ( F : A --> ( B X. C ) -> ( 1st o. F ) : A --> B )
Theorem	2ndcof 6113	Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
|- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C )
Theorem	xp1st 6114	Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A e. ( B X. C ) -> ( 1st ` A ) e. B )
Theorem	xp2nd 6115	Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )
Theorem	1stexg 6116	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( A e. V -> ( 1st ` A ) e. _V )
Theorem	2ndexg 6117	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( A e. V -> ( 2nd ` A ) e. _V )
Theorem	elxp6 6118	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5074. (Contributed by NM, 9-Oct-2004.)
|- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
Theorem	elxp7 6119	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5074. (Contributed by NM, 19-Aug-2006.)
|- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
Theorem	oprssdmm 6120*	Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
|- ( ( ph /\ u e. S ) -> E. v v e. u )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   &    |- ( ph -> Rel F )   =>    |- ( ph -> ( S X. S ) C_ dom F )
Theorem	eqopi 6121	Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
|- ( ( A e. ( V X. W ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) -> A = <. B , C >. )
Theorem	xp2 6122*	Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
|- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) }
Theorem	unielxp 6123	The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
|- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) )
Theorem	1st2nd2 6124	Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
|- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
Theorem	xpopth 6125	An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
|- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
Theorem	eqop 6126	Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
|- ( A e. ( V X. W ) -> ( A = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )
Theorem	eqop2 6127	Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
|- B e. _V   &    |- C e. _V   =>    |- ( A = <. B , C >. <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )
Theorem	op1steq 6128*	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.)
|- ( A e. ( V X. W ) -> ( ( 1st ` A ) = B <-> E. x A = <. B , x >. ) )
Theorem	2nd1st 6129	Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
|- ( A e. ( B X. C ) -> U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. )
Theorem	1st2nd 6130	Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
|- ( ( Rel B /\ A e. B ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
Theorem	1stdm 6131	The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
|- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )
Theorem	2ndrn 6132	The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
|- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R )
Theorem	1st2ndbr 6133	Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
|- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) )
Theorem	releldm2 6134*	Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
|- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )
Theorem	reldm 6135*	An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
|- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) )
Theorem	sbcopeq1a 6136	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2946 that avoids the existential quantifiers of copsexg 4205). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )
Theorem	csbopeq1a 6137	Equality theorem for substitution of a class A for an ordered pair <. x , y >. in B (analog of csbeq1a 3040). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( A = <. x , y >. -> [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B )
Theorem	dfopab2 6138*	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.)
|- { <. x , y >. | ph } = { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph }
Theorem	dfoprab3s 6139*	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.)
|- { <. <. x , y >. , z >. | ph } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }
Theorem	dfoprab3 6140*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
|- ( w = <. x , y >. -> ( ph <-> ps ) )   =>    |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } = { <. <. x , y >. , z >. | ps }
Theorem	dfoprab4 6141*	Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( w = <. x , y >. -> ( ph <-> ps ) )   =>    |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }
Theorem	dfoprab4f 6142*	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.)
|- F/ x ph   &    |- F/ y ph   &    |- ( w = <. x , y >. -> ( ph <-> ps ) )   =>    |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) }
Theorem	dfxp3 6143*	Define the cross product of three classes. Compare df-xp 4593. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
|- ( ( A X. B ) X. C ) = { <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) }
Theorem	elopabi 6144*	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
|- ( x = ( 1st ` A ) -> ( ph <-> ps ) )   &    |- ( y = ( 2nd ` A ) -> ( ps <-> ch ) )   =>    |- ( A e. { <. x , y >. | ph } -> ch )
Theorem	eloprabi 6145*	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.)
|- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) )   &    |- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) )   &    |- ( z = ( 2nd ` A ) -> ( ch <-> th ) )   =>    |- ( A e. { <. <. x , y >. , z >. | ph } -> th )
Theorem	mpomptsx 6146*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
Theorem	mpompts 6147*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
|- ( x e. A , y e. B |-> C ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
Theorem	dmmpossx 6148*	The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- dom F C_ U_ x e. A ( { x } X. B )
Theorem	fmpox 6149*	Functionality, domain and codomain of a class given by the maps-to notation, where B ( x ) is not constant but depends on x. (Contributed by NM, 29-Dec-2014.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )
Theorem	fmpo 6150*	Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. D <-> F : ( A X. B ) --> D )
Theorem	fnmpo 6151*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) )
Theorem	mpofvex 6152*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )
Theorem	fnmpoi 6153*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   =>    |- F Fn ( A X. B )
Theorem	dmmpo 6154*	Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   =>    |- dom F = ( A X. B )
Theorem	mpofvexi 6155*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   &    |- R e. _V   &    |- S e. _V   =>    |- ( R F S ) e. _V
Theorem	ovmpoelrn 6156*	An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
|- O = ( x e. A , y e. B |-> C )   =>    |- ( ( A. x e. A A. y e. B C e. M /\ X e. A /\ Y e. B ) -> ( X O Y ) e. M )
Theorem	dmmpoga 6157*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6154. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) )
Theorem	dmmpog 6158*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6154. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( C e. V -> dom F = ( A X. B ) )
Theorem	mpoexxg 6159*	Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( A e. R /\ A. x e. A B e. S ) -> F e. _V )
Theorem	mpoexg 6160*	Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( A e. R /\ B e. S ) -> F e. _V )
Theorem	mpoexga 6161*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.)
|- ( ( A e. V /\ B e. W ) -> ( x e. A , y e. B |-> C ) e. _V )
Theorem	mpoex 6162*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( x e. A , y e. B |-> C ) e. _V
Theorem	fnmpoovd 6163*	A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
|- ( ph -> M Fn ( A X. B ) )   &    |- ( ( i = a /\ j = b ) -> D = C )   &    |- ( ( ph /\ i e. A /\ j e. B ) -> D e. U )   &    |- ( ( ph /\ a e. A /\ b e. B ) -> C e. V )   =>    |- ( ph -> ( M = ( a e. A , b e. B |-> C ) <-> A. i e. A A. j e. B ( i M j ) = D ) )
Theorem	fmpoco 6164*	Composition of two functions. Variation of fmptco 5634 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C )   &    |- ( ph -> F = ( x e. A , y e. B |-> R ) )   &    |- ( ph -> G = ( z e. C |-> S ) )   &    |- ( z = R -> S = T )   =>    |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )
Theorem	oprabco 6165*	Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
|- ( ( x e. A /\ y e. B ) -> C e. D )   &    |- F = ( x e. A , y e. B |-> C )   &    |- G = ( x e. A , y e. B |-> ( H ` C ) )   =>    |- ( H Fn D -> G = ( H o. F ) )
Theorem	oprab2co 6166*	Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
|- ( ( x e. A /\ y e. B ) -> C e. R )   &    |- ( ( x e. A /\ y e. B ) -> D e. S )   &    |- F = ( x e. A , y e. B |-> <. C , D >. )   &    |- G = ( x e. A , y e. B |-> ( C M D ) )   =>    |- ( M Fn ( R X. S ) -> G = ( M o. F ) )
Theorem	df1st2 6167*	An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )
Theorem	df2nd2 6168*	An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )
Theorem	1stconst 6169	The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
|- ( B e. V -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A )
Theorem	2ndconst 6170	The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
|- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B )
Theorem	dfmpo 6171*	Alternate definition for the maps-to notation df-mpo 5830 (although it requires that C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- C e. _V   =>    |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }
Theorem	cnvf1olem 6172	Lemma for cnvf1o 6173. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )
Theorem	cnvf1o 6173*	Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )
Theorem	f2ndf 6174	The 2nd (second component of an ordered pair) function restricted to a function F is a function from F into the codomain of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
|- ( F : A --> B -> ( 2nd |` F ) : F --> B )
Theorem	fo2ndf 6175	The 2nd (second component of an ordered pair) function restricted to a function F is a function from F onto the range of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
|- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )
Theorem	f1o2ndf1 6176	The 2nd (second component of an ordered pair) function restricted to a one-to-one function F is a one-to-one function from F onto the range of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
|- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F )
Theorem	algrflem 6177	Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- B e. _V   &    |- C e. _V   =>    |- ( B ( F o. 1st ) C ) = ( F ` B )
Theorem	algrflemg 6178	Lemma for algrf 11926 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)
|- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )
Theorem	xporderlem 6179*	Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) }   =>    |- ( <. a , b >. T <. c , d >. <-> ( ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) )
Theorem	poxp 6180*	A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) }   =>    |- ( ( R Po A /\ S Po B ) -> T Po ( A X. B ) )
Theorem	spc2ed 6181*	Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
|- F/ x ch   &    |- F/ y ch   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( ch -> E. x E. y ps ) )
Theorem	cnvoprab 6182*	The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
|- F/ x ps   &    |- F/ y ps   &    |- ( a = <. x , y >. -> ( ps <-> ph ) )   &    |- ( ps -> a e. ( _V X. _V ) )   =>    |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }
Theorem	f1od2 6183*	Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
|- F = ( x e. A , y e. B |-> C )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. W )   &    |- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) )   &    |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( z e. D /\ ( x = I /\ y = J ) ) ) )   =>    |- ( ph -> F : ( A X. B ) -1-1-onto-> D )
Theorem	disjxp1 6184*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> Disj x e. A B )   =>    |- ( ph -> Disj x e. A ( B X. C ) )
Theorem	disjsnxp 6185*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- Disj j e. A ( { j } X. B )
2.6.16  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 5830) where the domain of the second argument depends on the domain of the first argument, especially when 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 "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 5900, ovmpox 5950 and fmpox 6149). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 6186*	Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4727. (Contributed by AV, 25-Oct-2020.)
|- F/_ x E   &    |- ( x = C -> B = E )   =>    |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
Theorem	mpoxopn0yelv 6187*	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.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> C )   =>    |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )
Theorem	mpoxopoveq 6188*	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.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )   =>    |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
Theorem	mpoxopovel 6189*	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.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )   =>    |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) <-> ( K e. V /\ N e. V /\ [. <. V , W >. / x ]. [. K / y ]. [. N / n ]. ph ) ) )
Theorem	rbropapd 6190*	Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )   &    |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )
Theorem	rbropap 6191*	Properties of a pair in a restricted binary relation M expressed as an ordered-pair class abstraction: M is the binary relation W restricted by the condition ps. (Contributed by AV, 31-Jan-2021.)
|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )   &    |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) )
2.6.17  Function transposition
Syntax	ctpos 6192	The transposition of a function.
class tpos F
Definition	df-tpos 6193*	Define the transposition of a function, which is a function G = tpos F satisfying G ( x , y ) = F ( y , x ). (Contributed by Mario Carneiro, 10-Sep-2015.)
|- tpos F = ( F o. ( x e. ( `' dom F u. { (/) } ) |-> U. `' { x } ) )
Theorem	tposss 6194	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F C_ G -> tpos F C_ tpos G )
Theorem	tposeq 6195	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F = G -> tpos F = tpos G )
Theorem	tposeqd 6196	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> F = G )   =>    |- ( ph -> tpos F = tpos G )
Theorem	tposssxp 6197	The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- tpos F C_ ( ( `' dom F u. { (/) } ) X. ran F )
Theorem	reltpos 6198	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- Rel tpos F
Theorem	brtpos2 6199	Value of the transposition at a pair <. A , B >.. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( B e. V -> ( Atpos F B <-> ( A e. ( `' dom F u. { (/) } ) /\ U. `' { A } F B ) ) )
Theorem	brtpos0 6200	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.)
|- ( A e. V -> ( (/)tpos F A <-> (/) F A ) )