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	1stexg 6401	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( A e. V -> ( 1st ` A ) e. _V )
Theorem	2ndexg 6402	Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( A e. V -> ( 2nd ` A ) e. _V )
Theorem	elxp6 6403	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5275. (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 6404	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5275. (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 6405*	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 6406	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 6407*	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 6408	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 6409	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 6410	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 6411	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 6412	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 6413*	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 6414	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 6415	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 6416	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 6417	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 6418	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 6419*	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 6420*	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 6421	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3061 that avoids the existential quantifiers of copsexg 4384). (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 6422	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 6423*	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 6424*	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 6425*	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 6426*	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 6427*	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 6428*	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 6429*	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 6430*	Define the cross product of three classes. Compare df-xp 4780. (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 6431*	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 6432*	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 6433*	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 6434*	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 6435*	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 6436*	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 6437*	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 6438*	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 6439*	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 6440*	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 6441*	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 6442*	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 6443*	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 6444*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6440. (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 6445*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6440. 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 6446*	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 6447*	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 6448*	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 6449*	Weak version of mpoex 6450 that holds without ax-coll 4246. 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 6450*	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 6451*	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 6452*	Composition of two functions. Variation of fmptco 5874 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 6453*	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 6454*	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 6455*	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 6456*	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 6457	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 6458	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 6459*	Alternate definition for the maps-to notation df-mpo 6090 (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 6460	Lemma for cnvf1o 6461. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )
Theorem	cnvf1o 6461*	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 6462	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 6463	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 6464	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 6465	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 6466	Lemma for algrf 12823 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 6467*	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 6468*	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 6469*	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 6470*	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 6471*	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 6472*	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 6473*	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 6474*	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 6477) are based on the Axiom of Union (usage of dmexg 5046), 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 6486). 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 6475	Extend class definition to include the support of functions.
class supp
Definition	df-supp 6476*	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 8910 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 6477*	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 6478	The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
|- ( Z e. W -> ( (/) supp Z ) = (/) )
Theorem	suppval1 6479*	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 6480*	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6481 assumes F is a set rather than its domain X, avoiding ax-coll 4246. (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 6481*	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 6482	An element of the support of a function with a given domain. This version of elsuppfn 6483 assumes F is a set rather than its domain X, avoiding ax-coll 4246. (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 6483	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 6484*	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 6485*	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 6486	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 6487	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 6488	Version of fsuppeq 6487 avoiding ax-coll 4246 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 6489	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 6490	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 6491	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 6492	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 6493*	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 6494	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 6495*	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 6496*	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 } )
Theorem	suppfnss 6497*	The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.)
|- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( F supp Z ) C_ ( G supp Z ) ) )
Theorem	funsssuppss 6498	The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
|- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )
Theorem	fczsupp0 6499	The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
|- ( ( B X. { Z } ) supp Z ) = (/)
Theorem	suppssdc 6500*	Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)
|- ( ph -> F : A --> B )   &    |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )   &    |- ( ph -> A. x e. A DECID x e. W )   =>    |- ( ph -> ( F supp Z ) C_ W )