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	eqop 6401	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 6402	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 6403*	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 6404	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 6405	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 6406	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 6407	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 6408	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 6409*	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 6410*	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 6411	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3061 that avoids the existential quantifiers of copsexg 4379). (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 6412	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 6413*	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 6414*	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 6415*	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 6416*	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 6417*	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 6418*	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 6419*	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 6420*	Define the cross product of three classes. Compare df-xp 4775. (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 6421*	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 6422*	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 6423*	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 6424*	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 6425*	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 6426*	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 6427*	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 6428*	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 6429*	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 6430*	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 6431*	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 6432*	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 6433*	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 6434*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6430. (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 6435*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6430. 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 6436*	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 6437*	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 6438*	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 6439*	Weak version of mpoex 6440 that holds without ax-coll 4241. 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 6440*	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 6441*	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 6442*	Composition of two functions. Variation of fmptco 5865 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 6443*	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 6444*	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 6445*	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 6446*	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 6447	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 6448	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 6449*	Alternate definition for the maps-to notation df-mpo 6080 (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 6450	Lemma for cnvf1o 6451. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )
Theorem	cnvf1o 6451*	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 6452	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 6453	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 6454	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 6455	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 6456	Lemma for algrf 12801 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 6457*	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 6458*	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 6459*	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 6460*	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 6461*	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 6462*	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 6463*	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 6464*	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 6467) are based on the Axiom of Union (usage of dmexg 5041), 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 6476). 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 6465	Extend class definition to include the support of functions.
class supp
Definition	df-supp 6466*	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 8900 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 6467*	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 6468	The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
|- ( Z e. W -> ( (/) supp Z ) = (/) )
Theorem	suppval1 6469*	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 6470*	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6471 assumes F is a set rather than its domain X, avoiding ax-coll 4241. (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 6471*	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 6472	An element of the support of a function with a given domain. This version of elsuppfn 6473 assumes F is a set rather than its domain X, avoiding ax-coll 4241. (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 6473	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 6474*	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 6475*	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 6476	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 6477	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 6478	Version of fsuppeq 6477 avoiding ax-coll 4241 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 6479	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 6480	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 6481	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 6482	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 6483*	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 6484	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 6485*	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 6486*	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 6487*	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 6488	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 6489	The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
|- ( ( B X. { Z } ) supp Z ) = (/)
Theorem	suppssdc 6490*	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 )
Theorem	suppssrst 6491*	A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
Theorem	suppssrgst 6492*	A function is zero outside its support. Version of suppssrst 6491 avoiding ax-coll 4241 by assuming F is a set rather than its domain A. (Contributed by SN, 5-May-2024.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> F e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
Theorem	suppssfvg 6493*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
|- ( ph -> ( ( x e. D |-> A ) supp Y ) C_ L )   &    |- ( ph -> ( F ` Y ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   &    |- ( ph -> Y e. U )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( ( x e. D |-> ( F ` A ) ) supp Z ) C_ L )
Theorem	suppofss1dcl 6494*	Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : A --> B )   &    |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )   &    |- ( ( ph /\ x e. B ) -> ( Z X x ) = Z )   =>    |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) )
Theorem	suppofss2dcl 6495*	Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : A --> B )   &    |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )   &    |- ( ( ph /\ x e. B ) -> ( x X Z ) = Z )   =>    |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) )
Theorem	suppcofn 6496	The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) (Revised by SN, 15-Sep-2023.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
Theorem	supp0cosupp0fn 6497	The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) )
Theorem	imacosuppfn 6498	The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )
2.6.17  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 6080) 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 6156, ovmpox 6207 and fmpox 6426). 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 6499*	Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4915. (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 6500*	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 ) )