P. 61 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fndmdif 6001*	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 6002	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	fndmdifeq0 6003	The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
|- ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> F = G ) )
Theorem	fndmin 6004*	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 6005	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 6006	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 6007	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 6008*	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 6009	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 6010	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 6011	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 6012	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 5667 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 6013	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 6012 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 6014*	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 6015*	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	fvimacnvALT 6016	Alternate proof of fvimacnv 6012, based on funimass3 6013. If funimass3 6013 is ever proved directly, as opposed to using funimacnv 5665 pointwise, then the proof of funimacnv 5665 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )
Theorem	elpreima 6017	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 6018	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 6019*	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 6020*	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	unpreima 6021	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
Theorem	inpreima 6022	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 6023	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
|- ( Fun F -> ( `' F " ( A \ B ) ) = ( ( `' F " A ) \ ( `' F " B ) ) )
Theorem	respreima 6024	The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( `' ( F |` B ) " A ) = ( ( `' F " A ) i^i B ) )
Theorem	iinpreima 6025*	Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
|- ( ( Fun F /\ A =/= (/) ) -> ( `' F " |^|_ x e. A B ) = |^|_ x e. A ( `' F " B ) )
Theorem	intpreima 6026*	Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
|- ( ( Fun F /\ A =/= (/) ) -> ( `' F " |^| A ) = |^|_ x e. A ( `' F " x ) )
Theorem	fimacnv 6027	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	fvn0ssdmfun 6028*	If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 5911. (Contributed by AV, 27-Jan-2020.)
|- ( A. a e. D ( F ` a ) =/= (/) -> ( D C_ dom F /\ Fun ( F |` D ) ) )
Theorem	fnopfv 6029	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 6030	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	nelrnfvne 6031	A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.)
|- ( ( Fun F /\ X e. dom F /\ Y e/ ran F ) -> ( F ` X ) =/= Y )
Theorem	fveqdmss 6032*	If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.)
|- D = dom B   =>    |- ( ( Fun B /\ (/) e/ ran B /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> D C_ dom A )
Theorem	fveqressseq 6033*	If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)
|- D = dom B   =>    |- ( ( Fun B /\ (/) e/ ran B /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> ( A |` D ) = B )
Theorem	fnfvelrn 6034	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 6035	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 6036	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 6037	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 6038	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 6039*	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 6040*	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 6041*	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 6042*	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 6043*	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	eldmrexrnb 6044*	For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 5597 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 5597 of the value of a function, ( F ` Y ) = (/) may mean that the value of F at Y is the empty set or that F is not defined at Y. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
|- ( ( Fun F /\ (/) e/ ran F ) -> ( Y e. dom F <-> E. x e. ran F x = ( F ` Y ) ) )
Theorem	fvcofneq 6045*	The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)
|- ( ( G Fn A /\ K Fn B ) -> ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) )
Theorem	ralrnmpt 6046*	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 6047*	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	f0cli 6048	Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)
|- F : A --> B   &    |- (/) e. B   =>    |- ( F ` C ) e. B
Theorem	dff2 6049	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )
Theorem	dff3 6050*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
|- ( F : A --> B <-> ( F C_ ( A X. B ) /\ A. x e. A E! y x F y ) )
Theorem	dff4 6051*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
|- ( F : A --> B <-> ( F C_ ( A X. B ) /\ A. x e. A E! y e. B x F y ) )
Theorem	dffo3 6052*	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 6053*	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 6054*	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	exfo 6055*	A relation equivalent to the existence of an onto mapping. The right-hand f is not necessarily a function. (Contributed by NM, 20-Mar-2007.)
|- ( E. f f : A -onto-> B <-> E. f ( A. x e. A E! y e. B x f y /\ A. x e. B E. y e. A y f x ) )
Theorem	foelrn 6056*	Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
|- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) )
Theorem	foco2 6057	If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
|- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> F : B -onto-> C )
Theorem	fmpt 6058*	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 6059*	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 6060*	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	fmptd 6061*	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	fmpt3d 6062*	Domain and co-domain 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 6063*	A version of fmptd 6061 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 6064*	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 6065	A function maps to a class to which all values belong. This version of ffnfv 6064 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 6066*	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	frnssb 6067*	A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.)
|- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) -> ( F : A --> W <-> F : A --> V ) )
Theorem	rnmptss 6068*	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 6069*	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 6070*	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	f1oresrab 6071*	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 6072*	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 6073*	Version of fmptco 6072 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 6074*	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	fcompt 6075*	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 6076	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 6077	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	fsn2 6078	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	fsng 6079	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	fsn2g 6080	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 6081	The Cartesian product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
Theorem	xpsn 6082	The Cartesian product of two singletons. (Contributed by NM, 4-Nov-2006.)
|- A e. _V   &    |- B e. _V   =>    |- ( { A } X. { B } ) = { <. A , B >. }
Theorem	f1o2sn 6083	A singleton with a nested ordered pair is a 1-1 function of the cartesian product of two singleton onto a singleton. (Contributed by AV, 15-Aug-2019.)
|- ( ( E e. V /\ X e. W ) -> { <. <. E , E >. , X >. } : ( { E } X. { E } ) -1-1-onto-> { X } )
Theorem	residpr 6084	Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
|- ( ( A e. V /\ B e. W ) -> ( _I |` { A , B } ) = { <. A , A >. , <. B , B >. } )
Theorem	dfmpt 6085	Alternate definition for the "maps to" notation df-mpt 4456 (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 6086	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	ressnop0 6087	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 6088	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 6089	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 6090	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 6091	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 6092	A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
|- ( ( F Fn A /\ B e. A ) -> ( F |` { B } ) = { <. B , ( F ` B ) >. } )
Theorem	funressn 6093	A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- ( Fun F -> ( F |` { B } ) C_ { <. B , ( F ` B ) >. } )
Theorem	fressnfv 6094	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	fvrnressn 6095	If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
|- ( X e. V -> ( ( F ` X ) e. ran ( F |` { X } ) -> ( F ` X ) e. ran F ) )
Theorem	fvressn 6096	The value of a function restricted to the singleton containing the argument equals the value of the function for this argument. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
|- ( X e. V -> ( ( F |` { X } ) ` X ) = ( F ` X ) )
Theorem	fvn0fvelrn 6097	If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
|- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F )
Theorem	fvconst 6098	The value of a constant function. (Contributed by NM, 30-May-1999.)
|- ( ( F : A --> { B } /\ C e. A ) -> ( F ` C ) = B )
Theorem	fnsnb 6099	A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)
|- A e. _V   =>    |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )
Theorem	fmptsn 6100*	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 ) )