P. 57 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5601-5700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mptrcl 5601*	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 5602*	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 5603*	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 5604*	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 5605*	Deduction version of fvmpt2 5602. (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 5606*	Alternate deduction version of fvmpt 5596, 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 5607*	Alternate deduction version of fvmpt 5596, 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 5608*	Alternate deduction version of fvmpt 5596, 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 5609*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5616. (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 5610*	Closed theorem form of fvmpt 5596. (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 5611*	Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5595 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 5612*	Deduction version of fvmpt 5596. (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 5613*	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 5614*	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 5615*	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 5616*	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 5617*	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 5618*	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 5619*	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 5620*	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 5616 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 5621*	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 5622*	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	fnmptfvd 5623*	A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)
|- ( ph -> M Fn A )   &    |- ( i = a -> D = C )   &    |- ( ( ph /\ i e. A ) -> D e. U )   &    |- ( ( ph /\ a e. A ) -> C e. V )   =>    |- ( ph -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = D ) )
Theorem	fndmdif 5624*	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 5625	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 5626*	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 5627	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 5628	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 5629	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 5630*	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 5631	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 5632	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 5633	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 5634	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 5296 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 5635	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 5634 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 5636*	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 5637*	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 5638	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 5639	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 5640*	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 5641*	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 5642*	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 5643*	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 5644	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
Theorem	inpreima 5645	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 5646	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
|- ( Fun F -> ( `' F " ( A \ B ) ) = ( ( `' F " A ) \ ( `' F " B ) ) )
Theorem	respreima 5647	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 5648	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 5649	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 5650	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 5651	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 5652	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 5653	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 5654	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 5655	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 5656*	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 5657*	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 5658*	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 5659*	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 5660*	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 5661*	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 5662*	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 5663	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )
Theorem	dff3im 5664*	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 5665*	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 5666*	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 5667*	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 5668*	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 5669*	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 5670*	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 5671*	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 5672*	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 5673*	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 5674*	Version of fmptd 5673 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 5675*	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 5676*	A version of fmptd 5673 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 5677*	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 5678	A function maps to a class to which all values belong. This version of ffnfv 5677 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 5679*	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 5680*	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 5681*	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 5682*	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 5683*	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 5683 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 5684*	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 5685*	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 5686*	Version of fmptco 5685 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 5687*	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 5688*	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 5689*	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 5690	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 5691	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 5692	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 5693	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	xpsng 5694	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 5695	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 5696	Alternate definition for the maps-to notation df-mpt 4068 (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 5697	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 5698	Alternate definition for the maps-to notation df-mpt 4068 (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 5699	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	ressnop0 5700	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 ) = (/) )