P. 59 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	funconstss 5801*	Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
|- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) = B <-> A C_ ( `' F " { B } ) ) )
Theorem	elpreima 5802	Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( F Fn A -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )
Theorem	fniniseg 5803	Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )
Theorem	fncnvima2 5804*	Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } )
Theorem	fniniseg2 5805*	Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) = B } )
Theorem	fnniniseg2 5806*	Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) =/= B } )
Theorem	unpreima 5807	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
Theorem	inpreima 5808	Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
|- ( Fun F -> ( `' F " ( A i^i B ) ) = ( ( `' F " A ) i^i ( `' F " B ) ) )
Theorem	difpreima 5809	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
|- ( Fun F -> ( `' F " ( A \ B ) ) = ( ( `' F " A ) \ ( `' F " B ) ) )
Theorem	respreima 5810	The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )
Theorem	fimacnv 5811	The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
|- ( F : A --> B -> ( `' F " B ) = A )
Theorem	fnopfv 5812	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
|- ( ( F Fn A /\ B e. A ) -> <. B , ( F ` B ) >. e. F )
Theorem	fvelrn 5813	A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
|- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )
Theorem	fnfvelrn 5814	A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
|- ( ( F Fn A /\ B e. A ) -> ( F ` B ) e. ran F )
Theorem	ffvelcdm 5815	A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
|- ( ( F : A --> B /\ C e. A ) -> ( F ` C ) e. B )
Theorem	ffvelcdmi 5816	A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
|- F : A --> B   =>    |- ( C e. A -> ( F ` C ) e. B )
Theorem	ffvelcdmda 5817	A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
|- ( ph -> F : A --> B )   =>    |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B )
Theorem	ffvelcdmd 5818	A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
|- ( ph -> F : A --> B )   &    |- ( ph -> C e. A )   =>    |- ( ph -> ( F ` C ) e. B )
Theorem	rexrn 5819*	Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
|- ( x = ( F ` y ) -> ( ph <-> ps ) )   =>    |- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) )
Theorem	ralrn 5820*	Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
|- ( x = ( F ` y ) -> ( ph <-> ps ) )   =>    |- ( F Fn A -> ( A. x e. ran F ph <-> A. y e. A ps ) )
Theorem	elrnrexdm 5821*	For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
|- ( Fun F -> ( Y e. ran F -> E. x e. dom F Y = ( F ` x ) ) )
Theorem	elrnrexdmb 5822*	For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
|- ( Fun F -> ( Y e. ran F <-> E. x e. dom F Y = ( F ` x ) ) )
Theorem	eldmrexrn 5823*	For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
|- ( Fun F -> ( Y e. dom F -> E. x e. ran F x = ( F ` Y ) ) )
Theorem	ralrnmpt 5824*	A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- F = ( x e. A |-> B )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A ch ) )
Theorem	rexrnmpt 5825*	A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- F = ( x e. A |-> B )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( A. x e. A B e. V -> ( E. y e. ran F ps <-> E. x e. A ch ) )
Theorem	dff2 5826	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )
Theorem	dff3im 5827*	Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
|- ( F : A --> B -> ( F C_ ( A X. B ) /\ A. x e. A E! y x F y ) )
Theorem	dff4im 5828*	Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
|- ( F : A --> B -> ( F C_ ( A X. B ) /\ A. x e. A E! y e. B x F y ) )
Theorem	dffo3 5829*	An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
Theorem	dffo4 5830*	Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A x F y ) )
Theorem	dffo5 5831*	Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x x F y ) )
Theorem	fmpt 5832*	Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A |-> C )   =>    |- ( A. x e. A C e. B <-> F : A --> B )
Theorem	f1ompt 5833*	Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
|- F = ( x e. A |-> C )   =>    |- ( F : A -1-1-onto-> B <-> ( A. x e. A C e. B /\ A. y e. B E! x e. A y = C ) )
Theorem	fmpti 5834*	Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. A |-> C )   &    |- ( x e. A -> C e. B )   =>    |- F : A --> B
Theorem	fvmptelcdm 5835*	The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> ( x e. A |-> B ) : A --> C )   =>    |- ( ( ph /\ x e. A ) -> B e. C )
Theorem	fmptd 5836*	Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
|- ( ( ph /\ x e. A ) -> B e. C )   &    |- F = ( x e. A |-> B )   =>    |- ( ph -> F : A --> C )
Theorem	fmpttd 5837*	Version of fmptd 5836 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.)
|- ( ( ph /\ x e. A ) -> B e. C )   =>    |- ( ph -> ( x e. A |-> B ) : A --> C )
Theorem	fmpt3d 5838*	Domain and codomain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. C )   =>    |- ( ph -> F : A --> C )
Theorem	fmptdf 5839*	A version of fmptd 5836 using bound-variable hypothesis instead of a distinct variable condition for ph. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B e. C )   &    |- F = ( x e. A |-> B )   =>    |- ( ph -> F : A --> C )
Theorem	ffnfv 5840*	A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
|- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
Theorem	ffnfvf 5841	A function maps to a class to which all values belong. This version of ffnfv 5840 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x F   =>    |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
Theorem	fnfvrnss 5842*	An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
|- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
Theorem	rnmptss 5843*	The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
|- F = ( x e. A |-> B )   =>    |- ( A. x e. A B e. C -> ran F C_ C )
Theorem	fmpt2d 5844*	Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
|- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C )   =>    |- ( ph -> F : A --> C )
Theorem	ffvresb 5845*	A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( Fun F -> ( ( F |` A ) : A --> B <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) )
Theorem	resflem 5846*	A lemma to bound the range of a restriction. The conclusion would also hold with ( X i^i Y ) in place of Y (provided x does not occur in X). If that stronger result is needed, it is however simpler to use the instance of resflem 5846 where ( X i^i Y ) is substituted for Y (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)
|- ( ph -> F : V --> X )   &    |- ( ph -> A C_ V )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) e. Y )   =>    |- ( ph -> ( F |` A ) : A --> Y )
Theorem	f1oresrab 5847*	Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
|- F = ( x e. A |-> C )   &    |- ( ph -> F : A -1-1-onto-> B )   &    |- ( ( ph /\ x e. A /\ y = C ) -> ( ch <-> ps ) )   =>    |- ( ph -> ( F |` { x e. A | ps } ) : { x e. A | ps } -1-1-onto-> { y e. B | ch } )
Theorem	fmptco 5848*	Composition of two functions expressed as ordered-pair class abstractions. If F has the equation ( x + 2 ) and G the equation ( 3 * z ) then ( G o. F ) has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
|- ( ( ph /\ x e. A ) -> R e. B )   &    |- ( ph -> F = ( x e. A |-> R ) )   &    |- ( ph -> G = ( y e. B |-> S ) )   &    |- ( y = R -> S = T )   =>    |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )
Theorem	fmptcof 5849*	Version of fmptco 5848 where ph needn't be distinct from x. (Contributed by NM, 27-Dec-2014.)
|- ( ph -> A. x e. A R e. B )   &    |- ( ph -> F = ( x e. A |-> R ) )   &    |- ( ph -> G = ( y e. B |-> S ) )   &    |- ( y = R -> S = T )   =>    |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )
Theorem	fmptcos 5850*	Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( ph -> A. x e. A R e. B )   &    |- ( ph -> F = ( x e. A |-> R ) )   &    |- ( ph -> G = ( y e. B |-> S ) )   =>    |- ( ph -> ( G o. F ) = ( x e. A |-> [_ R / y ]_ S ) )
Theorem	cofmpt 5851*	Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
|- ( ph -> F : C --> D )   &    |- ( ( ph /\ x e. A ) -> B e. C )   =>    |- ( ph -> ( F o. ( x e. A |-> B ) ) = ( x e. A |-> ( F ` B ) ) )
Theorem	fcompt 5852*	Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
|- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )
Theorem	fcoconst 5853	Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
|- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) )
Theorem	fsn 5854	A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( F : { A } --> { B } <-> F = { <. A , B >. } )
Theorem	fsng 5855	A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
|- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )
Theorem	fsn2 5856	A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
|- A e. _V   =>    |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )
Theorem	fsn2g 5857	A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
|- ( A e. V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
Theorem	xpsng 5858	The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
Theorem	xpsn 5859	The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)
|- A e. _V   &    |- B e. _V   =>    |- ( { A } X. { B } ) = { <. A , B >. }
Theorem	dfmpt 5860	Alternate definition for the maps-to notation df-mpt 4178 (although it requires that B be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
|- B e. _V   =>    |- ( x e. A |-> B ) = U_ x e. A { <. x , B >. }
Theorem	fnasrn 5861	A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- B e. _V   =>    |- ( x e. A |-> B ) = ran ( x e. A |-> <. x , B >. )
Theorem	dfmptg 5862	Alternate definition for the maps-to notation df-mpt 4178 (which requires that B be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)
|- ( A. x e. A B e. V -> ( x e. A |-> B ) = U_ x e. A { <. x , B >. } )
Theorem	fnasrng 5863	A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
|- ( A. x e. A B e. V -> ( x e. A |-> B ) = ran ( x e. A |-> <. x , B >. ) )
Theorem	funiun 5864*	A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)
|- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } )
Theorem	funopsn 5865*	If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 5407, as relsnopg 4859 is to relop 4910. (New usage is discouraged.)
|- X e. _V   &    |- Y e. _V   =>    |- ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )
Theorem	funop 5866*	An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 5407, as relsnopg 4859 is to relop 4910. (New usage is discouraged.)
|- X e. _V   &    |- Y e. _V   =>    |- ( Fun <. X , Y >. <-> E. a ( X = { a } /\ <. X , Y >. = { <. a , a >. } ) )
Theorem	fncofn 5867	Composition of a function with domain and a function as a function with domain. Generalization of fnco 5471. (Contributed by AV, 17-Sep-2024.)
|- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) )
Theorem	fcof 5868	Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5532. (Contributed by AV, 18-Sep-2024.)
|- ( ( F : A --> B /\ Fun G ) -> ( F o. G ) : ( `' G " A ) --> B )
Theorem	funopdmsn 5869	The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) (Avoid depending on this detail.)
|- G = <. X , Y >.   &    |- X e. V   &    |- Y e. W   =>    |- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B )
Theorem	ressnop0 5870	If A is not in C, then the restriction of a singleton of <. A , B >. to C is null. (Contributed by Scott Fenton, 15-Apr-2011.)
|- ( -. A e. C -> ( { <. A , B >. } |` C ) = (/) )
Theorem	fpr 5871	A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( A =/= B -> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } )
Theorem	fprg 5872	A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
|- ( ( ( A e. E /\ B e. F ) /\ ( C e. G /\ D e. H ) /\ A =/= B ) -> { <. A , C >. , <. B , D >. } : { A , B } --> { C , D } )
Theorem	ftpg 5873	A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { <. X , A >. , <. Y , B >. , <. Z , C >. } : { X , Y , Z } --> { A , B , C } )
Theorem	ftp 5874	A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- X e. _V   &    |- Y e. _V   &    |- Z e. _V   &    |- A =/= B   &    |- A =/= C   &    |- B =/= C   =>    |- { <. A , X >. , <. B , Y >. , <. C , Z >. } : { A , B , C } --> { X , Y , Z }
Theorem	fnressn 5875	A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
|- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )
Theorem	fressnfv 5876	The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
|- ( ( F Fn A /\ B e. A ) -> ( ( F |` { B } ) : { B } --> C <-> ( F ` B ) e. C ) )
Theorem	fvconst 5877	The value of a constant function. (Contributed by NM, 30-May-1999.)
|- ( ( F : A --> { B } /\ C e. A ) -> ( F ` C ) = B )
Theorem	fmptsn 5878*	Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } = ( x e. { A } |-> B ) )
Theorem	fmptap 5879*	Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- A e. _V   &    |- B e. _V   &    |- ( R u. { A } ) = S   &    |- ( x = A -> C = B )   =>    |- ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C )
Theorem	fmptapd 5880*	Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ph -> ( R u. { A } ) = S )   &    |- ( ( ph /\ x = A ) -> C = B )   =>    |- ( ph -> ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) )
Theorem	fmptpr 5881*	Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. Y )   &    |- ( ( ph /\ x = A ) -> E = C )   &    |- ( ( ph /\ x = B ) -> E = D )   =>    |- ( ph -> { <. A , C >. , <. B , D >. } = ( x e. { A , B } |-> E ) )
Theorem	fvresi 5882	The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
|- ( B e. A -> ( ( _I |` A ) ` B ) = B )
Theorem	fvunsng 5883	Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)
|- ( ( D e. V /\ B =/= D ) -> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) )
Theorem	fvsn 5884	The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
|- A e. _V   &    |- B e. _V   =>    |- ( { <. A , B >. } ` A ) = B
Theorem	fvsng 5885	The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
|- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B )
Theorem	fvsnun1 5886	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5887. (Contributed by NM, 23-Sep-2007.)
|- A e. _V   &    |- B e. _V   &    |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) )   =>    |- ( G ` A ) = B
Theorem	fvsnun2 5887	The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5886. (Contributed by NM, 23-Sep-2007.)
|- A e. _V   &    |- B e. _V   &    |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) )   =>    |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) )
Theorem	fnsnsplitss 5888	Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
|- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
Theorem	fsnunf 5889	Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- ( ( F : S --> T /\ ( X e. V /\ -. X e. S ) /\ Y e. T ) -> ( F u. { <. X , Y >. } ) : ( S u. { X } ) --> T )
Theorem	fsnunfv 5890	Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
|- ( ( X e. V /\ Y e. W /\ -. X e. dom F ) -> ( ( F u. { <. X , Y >. } ) ` X ) = Y )
Theorem	fsnunres 5891	Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- ( ( F Fn S /\ -. X e. S ) -> ( ( F u. { <. X , Y >. } ) |` S ) = F )
Theorem	funresdfunsnss 5892	Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
|- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
Theorem	fvpr1 5893	The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
|- A e. _V   &    |- C e. _V   =>    |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C )
Theorem	fvpr2 5894	The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
|- B e. _V   &    |- D e. _V   =>    |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D )
Theorem	fvpr1g 5895	The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
|- ( ( A e. V /\ C e. W /\ A =/= B ) -> ( { <. A , C >. , <. B , D >. } ` A ) = C )
Theorem	fvpr2g 5896	The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
|- ( ( B e. V /\ D e. W /\ A =/= B ) -> ( { <. A , C >. , <. B , D >. } ` B ) = D )
Theorem	fvtp1g 5897	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )
Theorem	fvtp2g 5898	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E )
Theorem	fvtp3g 5899	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( C e. V /\ F e. W ) /\ ( A =/= C /\ B =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F )
Theorem	fvtp1 5900	The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
|- A e. _V   &    |- D e. _V   =>    |- ( ( A =/= B /\ A =/= C ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )