P. 56 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fvco4 5501	Value of a composition. (Contributed by BJ, 7-Jul-2022.)
|- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( F ` u ) )
Theorem	fvopab3g 5502*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( x e. C -> E! y ph )   &    |- F = { <. x , y >. | ( x e. C /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) )
Theorem	fvopab3ig 5503*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( x e. C -> E* y ph )   &    |- F = { <. x , y >. | ( x e. C /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )
Theorem	fvmptss2 5504*	A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
|- ( x = D -> B = C )   &    |- F = ( x e. A |-> B )   =>    |- ( F ` D ) C_ C
Theorem	fvmptg 5505*	Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   =>    |- ( ( A e. D /\ C e. R ) -> ( F ` A ) = C )
Theorem	fvmpt 5506*	Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   &    |- C e. _V   =>    |- ( A e. D -> ( F ` A ) = C )
Theorem	fvmpts 5507*	Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. C |-> B )   =>    |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )
Theorem	fvmpt3 5508*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   &    |- ( x e. D -> B e. V )   =>    |- ( A e. D -> ( F ` A ) = C )
Theorem	fvmpt3i 5509*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   &    |- B e. _V   =>    |- ( A e. D -> ( F ` A ) = C )
Theorem	fvmptd 5510*	Deduction version of fvmpt 5506. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( ph -> F = ( x e. D |-> B ) )   &    |- ( ( ph /\ x = A ) -> B = C )   &    |- ( ph -> A e. D )   &    |- ( ph -> C e. V )   =>    |- ( ph -> ( F ` A ) = C )
Theorem	mptrcl 5511*	Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
|- F = ( x e. A |-> B )   =>    |- ( I e. ( F ` X ) -> X e. A )
Theorem	fvmpt2 5512*	Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
|- F = ( x e. A |-> B )   =>    |- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B )
Theorem	fvmptssdm 5513*	If all the values of the mapping are subsets of a class C, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
|- F = ( x e. A |-> B )   =>    |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )
Theorem	mptfvex 5514*	Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = ( x e. A |-> B )   =>    |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V )
Theorem	fvmpt2d 5515*	Deduction version of fvmpt2 5512. (Contributed by Thierry Arnoux, 8-Dec-2016.)
|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
Theorem	fvmptdf 5516*	Alternate deduction version of fvmpt 5506, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. D )   &    |- ( ( ph /\ x = A ) -> B e. V )   &    |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )   &    |- F/_ x F   &    |- F/ x ps   =>    |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )
Theorem	fvmptdv 5517*	Alternate deduction version of fvmpt 5506, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. D )   &    |- ( ( ph /\ x = A ) -> B e. V )   &    |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )   =>    |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )
Theorem	fvmptdv2 5518*	Alternate deduction version of fvmpt 5506, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. D )   &    |- ( ( ph /\ x = A ) -> B e. V )   &    |- ( ( ph /\ x = A ) -> B = C )   =>    |- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) )
Theorem	mpteqb 5519*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5526. (Contributed by Mario Carneiro, 14-Nov-2014.)
|- ( A. x e. A B e. V -> ( ( x e. A |-> B ) = ( x e. A |-> C ) <-> A. x e. A B = C ) )
Theorem	fvmptt 5520*	Closed theorem form of fvmpt 5506. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
|- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C )
Theorem	fvmptf 5521*	Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5505 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ x C   &    |- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   =>    |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )
Theorem	fvmptd3 5522*	Deduction version of fvmpt 5506. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- F = ( x e. D |-> B )   &    |- ( x = A -> B = C )   &    |- ( ph -> A e. D )   &    |- ( ph -> C e. V )   =>    |- ( ph -> ( F ` A ) = C )
Theorem	elfvmptrab1 5523*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )   &    |- ( X e. V -> [_ X / m ]_ M e. _V )   =>    |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
Theorem	elfvmptrab 5524*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- F = ( x e. V |-> { y e. M | ph } )   &    |- ( X e. V -> M e. _V )   =>    |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. M ) )
Theorem	fvopab6 5525*	Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
|- F = { <. x , y >. | ( ph /\ y = B ) }   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = A -> B = C )   =>    |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )
Theorem	eqfnfv 5526*	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
Theorem	eqfnfv2 5527*	Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
Theorem	eqfnfv3 5528*	Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) )
Theorem	eqfnfvd 5529*	Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn A )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) )   =>    |- ( ph -> F = G )
Theorem	eqfnfv2f 5530*	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5526 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
|- F/_ x F   &    |- F/_ x G   =>    |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
Theorem	eqfunfv 5531*	Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
|- ( ( Fun F /\ Fun G ) -> ( F = G <-> ( dom F = dom G /\ A. x e. dom F ( F ` x ) = ( G ` x ) ) ) )
Theorem	fvreseq 5532*	Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
|- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
Theorem	fndmdif 5533*	Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
|- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
Theorem	fndmdifcom 5534	The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
|- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = dom ( G \ F ) )
Theorem	fndmin 5535*	Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
|- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )
Theorem	fneqeql 5536	Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> dom ( F i^i G ) = A ) )
Theorem	fneqeql2 5537	Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A C_ dom ( F i^i G ) ) )
Theorem	fnreseql 5538	Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- ( ( F Fn A /\ G Fn A /\ X C_ A ) -> ( ( F |` X ) = ( G |` X ) <-> X C_ dom ( F i^i G ) ) )
Theorem	chfnrn 5539*	The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
|- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A )
Theorem	funfvop 5540	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
|- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
Theorem	funfvbrb 5541	Two ways to say that A is in the domain of F. (Contributed by Mario Carneiro, 1-May-2014.)
|- ( Fun F -> ( A e. dom F <-> A F ( F ` A ) ) )
Theorem	fvimacnvi 5542	A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
|- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )
Theorem	fvimacnv 5543	The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5209 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
|- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )
Theorem	funimass3 5544	A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5543 would be the special case of A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
|- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
Theorem	funimass5 5545*	A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
|- ( ( Fun F /\ A C_ dom F ) -> ( A C_ ( `' F " B ) <-> A. x e. A ( F ` x ) e. B ) )
Theorem	funconstss 5546*	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 5547	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 5548	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 5549*	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 5550*	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 5551*	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	rexsupp 5552*	Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
|- ( F Fn A -> ( E. x e. ( `' F " ( _V \ { Z } ) ) ph <-> E. x e. A ( ( F ` x ) =/= Z /\ ph ) ) )
Theorem	unpreima 5553	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
Theorem	inpreima 5554	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 5555	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
|- ( Fun F -> ( `' F " ( A \ B ) ) = ( ( `' F " A ) \ ( `' F " B ) ) )
Theorem	respreima 5556	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 5557	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 5558	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 5559	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 5560	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	ffvelrn 5561	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	ffvelrni 5562	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	ffvelrnda 5563	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	ffvelrnd 5564	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 5565*	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 5566*	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 5567*	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 5568*	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 5569*	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 5570*	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 5571*	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 5572	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )
Theorem	dff3im 5573*	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 5574*	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 5575*	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 5576*	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 5577*	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 5578*	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 5579*	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 5580*	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	fvmptelrn 5581*	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 5582*	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 5583*	Version of fmptd 5582 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 5584*	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 5585*	A version of fmptd 5582 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 5586*	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 5587	A function maps to a class to which all values belong. This version of ffnfv 5586 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 5588*	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 5589*	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 5590*	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 5591*	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 5592*	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 5592 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 5593*	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 5594*	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 5595*	Version of fmptco 5594 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 5596*	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 5597*	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 5598*	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 5599	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 5600	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 >. } )