P. 60 of Theorem List - Metamath Proof Explorer

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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fsnunres 5901	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	fvpr1 5902	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 5903	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 5904	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 5905	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 5906	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 5907	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 5908	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 5909	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( A e. V /\ D e. W ) /\ ( A =/= B /\ A =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` A ) = D )
Theorem	fvtp2g 5910	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( B e. V /\ E e. W ) /\ ( A =/= B /\ B =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` B ) = E )
Theorem	fvtp3g 5911	The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
|- ( ( ( C e. V /\ F e. W ) /\ ( A =/= C /\ B =/= C ) ) -> ( { <. A , D >. , <. B , E >. , <. C , F >. } ` C ) = F )
Theorem	fvconst2g 5912	The value of a constant function. (Contributed by NM, 20-Aug-2005.)
|- ( ( B e. D /\ C e. A ) -> ( ( A X. { B } ) ` C ) = B )
Theorem	fconst2g 5913	A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
|- ( B e. C -> ( F : A --> { B } <-> F = ( A X. { B } ) ) )
Theorem	fvconst2 5914	The value of a constant function. (Contributed by NM, 16-Apr-2005.)
|- B e. _V   =>    |- ( C e. A -> ( ( A X. { B } ) ` C ) = B )
Theorem	fconst2 5915	A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
|- B e. _V   =>    |- ( F : A --> { B } <-> F = ( A X. { B } ) )
Theorem	fconst5 5916	Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
|- ( ( F Fn A /\ A =/= (/) ) -> ( F = ( A X. { B } ) <-> ran F = { B } ) )
Theorem	fnpr 5917	Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) (Revised by NM, 10-Dec-2017.)
|- I e. _V   &    |- J e. _V   =>    |- ( I =/= J -> ( F Fn { I , J } <-> F = { <. I , ( F ` I ) >. , <. J , ( F ` J ) >. } ) )
Theorem	fnprOLD 5918	Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Obsolete version of fnpr 5917 as of 10-Dec-2017. (New usage is discouraged.) (Proof modification is discouraged.)
|- I e. A   &    |- J e. B   =>    |- ( I =/= J -> ( F Fn { I , J } <-> F = { <. I , ( F ` I ) >. , <. J , ( F ` J ) >. } ) )
Theorem	fnsuppres 5919	Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.)
|- ( ( F Fn ( A u. B ) /\ ( A i^i B ) = (/) /\ Z e. V ) -> ( ( `' F " ( _V \ { Z } ) ) C_ A <-> ( F |` B ) = ( B X. { Z } ) ) )
Theorem	fnsuppeq0 5920	The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.)
|- ( ( F Fn A /\ Z e. V ) -> ( ( `' F " ( _V \ { Z } ) ) = (/) <-> F = ( A X. { Z } ) ) )
Theorem	fconstfv 5921*	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5915. (Contributed by NM, 27-Aug-2004.)
|- ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) = B ) )
Theorem	fconst3 5922	Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
|- ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) )
Theorem	fconst4 5923	Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
|- ( F : A --> { B } <-> ( F Fn A /\ ( `' F " { B } ) = A ) )
Theorem	resfunexg 5924	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
|- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )
Theorem	resfunexgALT 5925	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5924 but requires ax-pow 4345. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )
Theorem	cofunexg 5926	Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
|- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )
Theorem	cofunex2g 5927	Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
|- ( ( A e. V /\ Fun `' B ) -> ( A o. B ) e. _V )
Theorem	fnex 5928	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5924. See fnexALT 5929 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( F Fn A /\ A e. B ) -> F e. _V )
Theorem	fnexALT 5929	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5497. This version of fnex 5928 uses ax-pow 4345, whereas fnex 5928 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( F Fn A /\ A e. B ) -> F e. _V )
Theorem	funex 5930	If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5928. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
|- ( ( Fun F /\ dom F e. B ) -> F e. _V )
Theorem	opabex 5931*	Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
|- A e. _V   &    |- ( x e. A -> E* y ph )   =>    |- { <. x , y >. | ( x e. A /\ ph ) } e. _V
Theorem	mptexg 5932*	If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( A e. V -> ( x e. A |-> B ) e. _V )
Theorem	mptex 5933*	If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
|- A e. _V   =>    |- ( x e. A |-> B ) e. _V
Theorem	funrnex 5934	If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5930. (Contributed by NM, 11-Nov-1995.)
|- ( dom F e. B -> ( Fun F -> ran F e. _V ) )
Theorem	zfrep6 5935*	A version of the Axiom of Replacement. Normally ph would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4298 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 4288. (Contributed by NM, 10-Oct-2003.)
|- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph )
Theorem	fex 5936	If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
|- ( ( F : A --> B /\ A e. C ) -> F e. _V )
Theorem	fornex 5937	If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
|- ( A e. C -> ( F : A -onto-> B -> B e. _V ) )
Theorem	f1dmex 5938	If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4288. (Contributed by NM, 4-Sep-2004.)
|- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V )
Theorem	eufnfv 5939*	A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
|- A e. _V   &    |- B e. _V   =>    |- E! f ( f Fn A /\ A. x e. A ( f ` x ) = B )
Theorem	funfvima 5940	A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
|- ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )
Theorem	funfvima2 5941	A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
|- ( ( Fun F /\ A C_ dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )
Theorem	funfvima3 5942	A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
|- ( ( Fun F /\ F C_ G ) -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) )
Theorem	fnfvima 5943	The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
|- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) )
Theorem	rexima 5944*	Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
|- ( x = ( F ` y ) -> ( ph <-> ps ) )   =>    |- ( ( F Fn A /\ B C_ A ) -> ( E. x e. ( F " B ) ph <-> E. y e. B ps ) )
Theorem	ralima 5945*	Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
|- ( x = ( F ` y ) -> ( ph <-> ps ) )   =>    |- ( ( F Fn A /\ B C_ A ) -> ( A. x e. ( F " B ) ph <-> A. y e. B ps ) )
Theorem	idref 5946*	TODO: This is the same as issref 5214 (which has a much longer proof). Should we replace issref 5214 with this one? - NM 9-May-2016. Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)
|- ( ( _I |` A ) C_ R <-> A. x e. A x R x )
Theorem	fvclss 5947*	Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
|- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } )
Theorem	fvclex 5948*	Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
|- F e. _V   =>    |- { y | E. x y = ( F ` x ) } e. _V
Theorem	fvresex 5949*	Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)
|- A e. _V   =>    |- { y | E. x y = ( ( F |` A ) ` x ) } e. _V
Theorem	abrexex 5950*	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5932, funex 5930, fnex 5928, resfunexg 5924, and funimaexg 5497. See also abrexex2 5968. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- A e. _V   =>    |- { y | E. x e. A y = B } e. _V
Theorem	abrexexg 5951*	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in B. The antecedent assures us that A is a set. (Contributed by NM, 3-Nov-2003.)
|- ( A e. V -> { y | E. x e. A y = B } e. _V )
Theorem	elabrex 5952*	Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
|- B e. _V   =>    |- ( x e. A -> B e. { y | E. x e. A y = B } )
Theorem	abrexco 5953*	Composition of two image maps C ( y ) and B ( w ). (Contributed by NM, 27-May-2013.)
|- B e. _V   &    |- ( y = B -> C = D )   =>    |- { x | E. y e. { z | E. w e. A z = B } x = C } = { x | E. w e. A x = D }
Theorem	iunexg 5954*	The existence of an indexed union. x is normally a free-variable parameter in B. (Contributed by NM, 23-Mar-2006.)
|- ( ( A e. V /\ A. x e. A B e. W ) -> U_ x e. A B e. _V )
Theorem	abrexex2g 5955*	Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( A e. V /\ A. x e. A { y | ph } e. W ) -> { y | E. x e. A ph } e. _V )
Theorem	opabex3d 5956*	Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
|- ( ph -> A e. _V )   &    |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )   =>    |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )
Theorem	opabex3 5957*	Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. _V   &    |- ( x e. A -> { y | ph } e. _V )   =>    |- { <. x , y >. | ( x e. A /\ ph ) } e. _V
Theorem	iunex 5958*	The existence of an indexed union. x is normally a free-variable parameter in the class expression substituted for B, which can be read informally as B ( x ). (Contributed by NM, 13-Oct-2003.)
|- A e. _V   &    |- B e. _V   =>    |- U_ x e. A B e. _V
Theorem	imaiun 5959*	The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( A " U_ x e. B C ) = U_ x e. B ( A " C )
Theorem	imauni 5960*	The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
|- ( A " U. B ) = U_ x e. B ( A " x )
Theorem	fniunfv 5961*	The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
|- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )
Theorem	funiunfv 5962*	The indexed union of a function's values is the union of its image under the index class. Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to F Fn A, the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) )
Theorem	funiunfvf 5963*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5962 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)
|- F/_ x F   =>    |- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) )
Theorem	eluniima 5964*	Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
|- ( Fun F -> ( B e. U. ( F " A ) <-> E. x e. A B e. ( F ` x ) ) )
Theorem	elunirn 5965*	Membership in the union of the range of a function. See elunirnALT 5967 for alternate proof. (Contributed by NM, 24-Sep-2006.)
|- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )
Theorem	fnunirn 5966*	Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( F Fn I -> ( A e. U. ran F <-> E. x e. I A e. ( F ` x ) ) )
Theorem	elunirnALT 5967*	Membership in the union of the range of a function, proved directly. Unlike elunirn 5965, it doesn't appeal to ndmfv 5722 (via funiunfv 5962). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )
Theorem	abrexex2 5968*	Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 5950. (Contributed by NM, 12-Sep-2004.)
|- A e. _V   &    |- { y | ph } e. _V   =>    |- { y | E. x e. A ph } e. _V
Theorem	abexssex 5969*	Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in ph. (Contributed by NM, 29-Jul-2006.)
|- A e. _V   &    |- { y | ph } e. _V   =>    |- { y | E. x ( x C_ A /\ ph ) } e. _V
Theorem	abexex 5970*	A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
|- A e. _V   &    |- ( ph -> x e. A )   &    |- { y | ph } e. _V   =>    |- { y | E. x ph } e. _V
Theorem	dff13 5971*	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
Theorem	f1veqaeq 5972	If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
|- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) )
Theorem	dff13f 5973*	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
|- F/_ x F   &    |- F/_ y F   =>    |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
Theorem	f1mpt 5974*	Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- F = ( x e. A |-> C )   &    |- ( x = y -> C = D )   =>    |- ( F : A -1-1-> B <-> ( A. x e. A C e. B /\ A. x e. A A. y e. A ( C = D -> x = y ) ) )
Theorem	f1fveq 5975	Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
|- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) = ( F ` D ) <-> C = D ) )
Theorem	f1elima 5976	Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( F : A -1-1-> B /\ X e. A /\ Y C_ A ) -> ( ( F ` X ) e. ( F " Y ) <-> X e. Y ) )
Theorem	f1imass 5977	Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
Theorem	f1imaeq 5978	Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )
Theorem	f1imapss 5979	Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C. ( F " D ) <-> C C. D ) )
Theorem	dff1o6 5980*	A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
Theorem	f1ocnvfv1 5981	The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
|- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( `' F ` ( F ` C ) ) = C )
Theorem	f1ocnvfv2 5982	The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( F ` ( `' F ` C ) ) = C )
Theorem	f1ocnvfv 5983	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
Theorem	f1ocnvfvb 5984	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
|- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D <-> ( `' F ` D ) = C ) )
Theorem	f1ocnvdm 5985	The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) e. A )
Theorem	f1ocnvfvrneq 5986	If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
|- ( ( F : A -1-1-> B /\ ( C e. ran F /\ D e. ran F ) ) -> ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) )
Theorem	fcof1 5987	An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )
Theorem	fcofo 5988	An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
|- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A -onto-> B )
Theorem	cbvfo 5989*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
|- ( ( F ` x ) = y -> ( ph <-> ps ) )   =>    |- ( F : A -onto-> B -> ( A. x e. A ph <-> A. y e. B ps ) )
Theorem	cbvexfo 5990*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
|- ( ( F ` x ) = y -> ( ph <-> ps ) )   =>    |- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) )
Theorem	cocan1 5991	An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
|- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( ( F o. H ) = ( F o. K ) <-> H = K ) )
Theorem	cocan2 5992	A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
|- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( ( H o. F ) = ( K o. F ) <-> H = K ) )
Theorem	fcof1o 5993	Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( ( F : A --> B /\ G : B --> A ) /\ ( ( F o. G ) = ( _I |` B ) /\ ( G o. F ) = ( _I |` A ) ) ) -> ( F : A -1-1-onto-> B /\ `' F = G ) )
Theorem	foeqcnvco 5994	Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
|- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> ( F = G <-> ( F o. `' G ) = ( _I |` B ) ) )
Theorem	f1eqcocnv 5995	Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
|- ( ( F : A -1-1-> B /\ G : A -1-1-> B ) -> ( F = G <-> ( `' F o. G ) = ( _I |` A ) ) )
Theorem	fveqf1o 5996	Given a bijection F, produce another bijection G which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)
|- G = ( F o. ( ( _I |` ( A \ { C , ( `' F ` D ) } ) ) u. { <. C , ( `' F ` D ) >. , <. ( `' F ` D ) , C >. } ) )   =>    |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( G : A -1-1-onto-> B /\ ( G ` C ) = D ) )
Theorem	fliftrel 5997*	F, a function lift, is a subset of R X. S. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. A , B >. )   &    |- ( ( ph /\ x e. X ) -> A e. R )   &    |- ( ( ph /\ x e. X ) -> B e. S )   =>    |- ( ph -> F C_ ( R X. S ) )
Theorem	fliftel 5998*	Elementhood in the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. A , B >. )   &    |- ( ( ph /\ x e. X ) -> A e. R )   &    |- ( ( ph /\ x e. X ) -> B e. S )   =>    |- ( ph -> ( C F D <-> E. x e. X ( C = A /\ D = B ) ) )
Theorem	fliftel1 5999*	Elementhood in the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. A , B >. )   &    |- ( ( ph /\ x e. X ) -> A e. R )   &    |- ( ( ph /\ x e. X ) -> B e. S )   =>    |- ( ( ph /\ x e. X ) -> A F B )
Theorem	fliftcnv 6000*	Converse of the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. A , B >. )   &    |- ( ( ph /\ x e. X ) -> A e. R )   &    |- ( ( ph /\ x e. X ) -> B e. S )   =>    |- ( ph -> `' F = ran ( x e. X |-> <. B , A >. ) )