P. 65 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1stcof 6401	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 6402	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 6403	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 6404	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 6405	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( A e. V -> ( 1st ` A ) e. _V )
Theorem	2ndexg 6406	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( A e. V -> ( 2nd ` A ) e. _V )
Theorem	elxp6 6407	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5278. (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 6408	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5278. (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 6409*	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 6410	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 6411*	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 6412	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 6413	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 6414	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 6415	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 6416	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 6417*	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 6418	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 6419	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 6420	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 6421	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 6422	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 6423*	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 6424*	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 6425	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3061 that avoids the existential quantifiers of copsexg 4386). (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 6426	Equality theorem for substitution of a class A for an ordered pair <. x , y >. in B (analog of csbeq1a 3156). (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 6427*	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 6428*	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 6429*	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 6430*	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 6431*	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	opabex2 6432*	Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ( ph /\ ps ) -> x e. A )   &    |- ( ( ph /\ ps ) -> y e. B )   =>    |- ( ph -> { <. x , y >. | ps } e. _V )
Theorem	opabn1stprc 6433*	An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.)
|- ( E. y ph -> { <. x , y >. | ph } e/ _V )
Theorem	dfxp3 6434*	Define the cross product of three classes. Compare df-xp 4782. (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 6435*	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 6436*	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 6437*	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 6438*	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 6439*	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 6440*	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 6441*	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 6442*	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	fnmpoi 6443*	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 6444*	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	mpofvex 6445*	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	mpofvexi 6446*	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 6447*	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 6448*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6444. (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 6449*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6444. 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 6450*	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 6451*	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 6452*	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	mpoexw 6453*	Weak version of mpoex 6454 that holds without ax-coll 4248. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
|- A e. _V   &    |- B e. _V   &    |- D e. _V   &    |- A. x e. A A. y e. B C e. D   =>    |- ( x e. A , y e. B |-> C ) e. _V
Theorem	mpoex 6454*	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 6455*	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 6456*	Composition of two functions. Variation of fmptco 5877 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 6457*	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 6458*	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 6459*	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 6460*	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 6461	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 6462	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 6463*	Alternate definition for the maps-to notation df-mpo 6093 (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 6464	Lemma for cnvf1o 6465. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )
Theorem	cnvf1o 6465*	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 6466	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 6467	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 6468	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 6469	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 6470	Lemma for algrf 12848 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 6471*	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 6472*	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 6473*	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 6474*	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 6475*	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 6476*	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 6477*	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 )
Theorem	elmpom 6478*	If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( D e. F -> E. z z e. A )
2.6.16  The support of functions In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 6481) are based on the Axiom of Union (usage of dmexg 5048), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression ( `' R " ( _V \ { Z } ) ) (see suppimacnvfn 6490). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now).
Syntax	csupp 6479	Extend class definition to include the support of functions.
class supp
Definition	df-supp 6480*	Define the support of a function against a "zero" value. The support of a function is the subset of its domain which is mapped to a value which is not equal to a designed value called the zero value. Note that this definition uses not equal rather than being in terms of an apartness relation (df-ap 8918 or any other apartness relation), and thus is sometimes called "support" rather than "strong support". It is therefore probably most useful when the function has a codomain which has decidable equality and contains the zero value. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
|- supp = ( x e. _V , z e. _V |-> { i e. dom x | ( x " { i } ) =/= { z } } )
Theorem	suppval 6481*	The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
|- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
Theorem	supp0 6482	The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
|- ( Z e. W -> ( (/) supp Z ) = (/) )
Theorem	suppval1 6483*	The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
|- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X ` i ) =/= Z } )
Theorem	suppvalfng 6484*	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6485 assumes F is a set rather than its domain X, avoiding ax-coll 4248. (Contributed by SN, 5-Aug-2024.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )
Theorem	suppvalfn 6485*	The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )
Theorem	elsuppfng 6486	An element of the support of a function with a given domain. This version of elsuppfn 6487 assumes F is a set rather than its domain X, avoiding ax-coll 4248. (Contributed by SN, 5-Aug-2024.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )
Theorem	elsuppfn 6487	An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )
Theorem	fvdifsuppst 6488*	Function value is zero outside of its support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
|- ( ph -> F : A --> B )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. x e. B A. y e. B STAB x = y )   &    |- ( ph -> Z e. B )   &    |- ( ph -> X e. ( A \ ( F supp Z ) ) )   =>    |- ( ph -> ( F ` X ) = Z )
Theorem	cnvimadfsn 6489*	The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
|- ( `' R " ( _V \ { Z } ) ) = { x | E. y ( x R y /\ y =/= Z ) }
Theorem	suppimacnvfn 6490	Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
Theorem	fsuppeq 6491	Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
|- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )
Theorem	fsuppeqg 6492	Version of fsuppeq 6491 avoiding ax-coll 4248 by assuming F is a set rather than its domain I. (Contributed by SN, 30-Jul-2024.)
|- ( ( F e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )
Theorem	suppssdmg 6493	The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
|- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) C_ dom F )
Theorem	suppsnopdc 6494	The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
|- F = { <. X , Y >. }   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. W )   &    |- ( ph -> Z e. U )   &    |- ( ph -> DECID Y = Z )   =>    |- ( ph -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )
Theorem	fvn0elsupp 6495	If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
|- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> X e. ( G supp (/) ) )
Theorem	fvn0elsuppb 6496	The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
|- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) <-> X e. ( G supp (/) ) ) )
Theorem	rexsupp 6497*	Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( E. x e. ( F supp Z ) ph <-> E. x e. X ( ( F ` x ) =/= Z /\ ph ) ) )
Theorem	ressuppss 6498	The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
|- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) )
Theorem	mptsuppdifd 6499*	The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
|- F = ( x e. A |-> B )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. W )   =>    |- ( ph -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } )
Theorem	mptsuppd 6500*	The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
|- F = ( x e. A |-> B )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. W )   &    |- ( ( ph /\ x e. A ) -> B e. U )   =>    |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )