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	fvelrn 6001	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 6002	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 6003*	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 6004*	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 6005	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 6006	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 6007	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 6008	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 6009	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 6010*	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 6011*	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 6012*	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 6013*	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 6014*	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 6015*	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 5576 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 5576 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 6016*	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 6017*	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 6018*	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 6019	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 6020	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )
Theorem	dff3 6021*	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 6022*	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 6023*	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 6024*	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 6025*	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 6026*	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 6027*	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 6028	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 6029*	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 6030*	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 6031*	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 6032*	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 6033*	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 6034*	A version of fmptd 6032 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 6035*	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 6036	A function maps to a class to which all values belong. This version of ffnfv 6035 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 6037*	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 6038*	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 6039*	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 6040*	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 6041*	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 6042*	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 6043*	Version of fmptco 6042 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 6044*	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 6045*	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 6046	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 6047	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 6048	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 6049	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 6050	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 6051	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 6052	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 6053	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 6054	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 6055	Alternate definition for the "maps to" notation df-mpt 4454 (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 6056	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 6057	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 6058	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 6059	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 6060	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 6061	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 6062	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 6063	A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- ( Fun F -> ( F |` { B } ) C_ { <. B , ( F ` B ) >. } )
Theorem	fressnfv 6064	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 6065	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 6066	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 6067	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 6068	The value of a constant function. (Contributed by NM, 30-May-1999.)
|- ( ( F : A --> { B } /\ C e. A ) -> ( F ` C ) = B )
Theorem	fnsnb 6069	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 6070*	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	fmptsng 6071*	Express a singleton function in maps-to notation. Version of fmptsn 6070 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)
|- ( x = A -> B = C )   =>    |- ( ( A e. V /\ C e. W ) -> { <. A , C >. } = ( x e. { A } |-> B ) )
Theorem	fmptsnd 6072*	Express a singleton function in maps-to notation. Deduction form of fmptsng 6071. (Contributed by AV, 4-Aug-2019.)
|- ( ( ph /\ x = A ) -> B = C )   &    |- ( ph -> A e. V )   &    |- ( ph -> C e. W )   =>    |- ( ph -> { <. A , C >. } = ( x e. { A } |-> B ) )
Theorem	fmptap 6073*	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 6074*	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 6075*	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 6076	The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
|- ( B e. A -> ( ( _I |` A ) ` B ) = B )
Theorem	fninfp 6077*	Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( F Fn A -> dom ( F i^i _I ) = { x e. A | ( F ` x ) = x } )
Theorem	fnelfp 6078	Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F i^i _I ) <-> ( F ` X ) = X ) )
Theorem	fndifnfp 6079*	Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( F Fn A -> dom ( F \ _I ) = { x e. A | ( F ` x ) =/= x } )
Theorem	fnelnfp 6080	Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F \ _I ) <-> ( F ` X ) =/= X ) )
Theorem	fnnfpeq0 6081	A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
|- ( F Fn A -> ( dom ( F \ _I ) = (/) <-> F = ( _I |` A ) ) )
Theorem	fvunsn 6082	Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
|- ( B =/= D -> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) )
Theorem	fvsn 6083	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 6084	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 6085	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6086. (Contributed by NM, 23-Sep-2007.)
|- A e. _V   &    |- B e. _V   &    |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) )   =>    |- ( G ` A ) = B
Theorem	fvsnun2 6086	The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6085. (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	fnsnsplit 6087	Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)
|- ( ( F Fn A /\ X e. A ) -> F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
Theorem	fsnunf 6088	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	fsnunf2 6089	Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- ( ( F : ( S \ { X } ) --> T /\ X e. S /\ Y e. T ) -> ( F u. { <. X , Y >. } ) : S --> T )
Theorem	fsnunfv 6090	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 6091	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	funresdfunsn 6092	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 the function itself. (Contributed by AV, 2-Dec-2018.)
|- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F )
Theorem	fvpr1 6093	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 6094	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 6095	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 6096	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	fvtp1 6097	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 )
Theorem	fvtp2 6098	The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
|- B e. _V   &    |- E e. _V   =>    |- ( ( A =/= B /\ B =/= C ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E )
Theorem	fvtp3 6099	The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
|- C e. _V   &    |- F e. _V   =>    |- ( ( A =/= C /\ B =/= C ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F )
Theorem	fvtp1g 6100	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 )