P. 60 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fvsn 5901	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 5902	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 5903	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5904. (Contributed by NM, 23-Sep-2007.)
|- A e. _V   &    |- B e. _V   &    |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) )   =>    |- ( G ` A ) = B
Theorem	fvsnun2 5904	The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5903. (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	fnsnsplitss 5905	Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
|- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
Theorem	fsnunf 5906	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	fsnunfv 5907	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 5908	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	funresdfunsnss 5909	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 a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
|- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
Theorem	fvpr1 5910	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 5911	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 5912	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 5913	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	fvtp1g 5914	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 5915	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 5916	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	fvtp1 5917	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 5918	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 5919	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	fvconst2g 5920	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 5921	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 5922	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 5923	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	fconstfvm 5924*	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5923. (Contributed by Jim Kingdon, 8-Jan-2019.)
|- ( E. y y e. A -> ( F : A --> { B } <-> ( F Fn A /\ A. x e. A ( F ` x ) = B ) ) )
Theorem	fconst3m 5925*	Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
|- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ A C_ ( `' F " { B } ) ) ) )
Theorem	fconst4m 5926*	Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
|- ( E. x x e. A -> ( F : A --> { B } <-> ( F Fn A /\ ( `' F " { B } ) = A ) ) )
Theorem	resfunexg 5927	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	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 5927. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( F Fn A /\ A e. B ) -> F e. _V )
Theorem	fndmexb 5929	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
|- ( F Fn A -> ( A e. _V <-> F e. _V ) )
Theorem	fdmexb 5930	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
|- ( F : A --> B -> ( A e. _V <-> F e. _V ) )
Theorem	funex 5931	If the domain of a function exists, so does 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 5932*	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 5933*	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 5934*	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	mptexd 5935*	If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 5933. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. V )   =>    |- ( ph -> ( x e. A |-> B ) e. _V )
Theorem	mptrabex 5936*	If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
|- A e. _V   =>    |- ( x e. { y e. A | ph } |-> B ) e. _V
Theorem	fex 5937	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	fexd 5938	If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   &    |- ( ph -> A e. C )   =>    |- ( ph -> F 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	fnfvimad 5944	A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> F Fn A )   &    |- ( ph -> B e. A )   &    |- ( ph -> B e. C )   =>    |- ( ph -> ( F ` B ) e. ( F " C ) )
Theorem	dfimafnf 5945*	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
|- F/_ x A   &    |- F/_ x F   =>    |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )
Theorem	resfvresima 5946	The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.)
|- ( ph -> Fun F )   &    |- ( ph -> S C_ dom F )   &    |- ( ph -> X e. S )   =>    |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )
Theorem	foima2 5947*	Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5615). (Contributed by BJ, 6-Jul-2022.)
|- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) )
Theorem	foelrn 5948*	Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.)
|- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) )
Theorem	foco2 5949	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	rexima 5950*	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 5951*	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 5952*	TODO: This is the same as issref 5165 (which has a much longer proof). Should we replace issref 5165 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	elabrex 5953*	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	elabrexg 5954*	Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( x e. A /\ B e. V ) -> B e. { y | E. x e. A y = B } )
Theorem	abrexco 5955*	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	imaiun 5956*	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 5957*	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 5958*	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	funiunfvdm 5959*	The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5958. (Contributed by Jim Kingdon, 10-Jan-2019.)
|- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )
Theorem	funiunfvdmf 5960*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5959 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
|- F/_ x F   =>    |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )
Theorem	eluniimadm 5961*	Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
|- ( F Fn A -> ( B e. U. ( F " A ) <-> E. x e. A B e. ( F ` x ) ) )
Theorem	elunirn 5962*	Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
|- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )
Theorem	fnunirn 5963*	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	dff13 5964*	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 5965	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 5966*	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 5967*	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 5968	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 5969	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 5970	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 5971	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	dff1o6 5972*	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 5973	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 5974	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 5975	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 5976	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 5977	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 5978	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 5979	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 5980	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 5981*	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 5982*	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 5983	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 5984	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 5985	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 5986	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 5987	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	fliftrel 5988*	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 5989*	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 5990*	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 5991*	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 >. ) )
Theorem	fliftfun 5992*	The function F is the unique function defined by F ` A = B, provided that the well-definedness condition holds. (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 )   &    |- ( x = y -> A = C )   &    |- ( x = y -> B = D )   =>    |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( A = C -> B = D ) ) )
Theorem	fliftfund 5993*	The function F is the unique function defined by F ` A = B, provided that the well-definedness condition holds. (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 )   &    |- ( x = y -> A = C )   &    |- ( x = y -> B = D )   &    |- ( ( ph /\ ( x e. X /\ y e. X /\ A = C ) ) -> B = D )   =>    |- ( ph -> Fun F )
Theorem	fliftfuns 5994*	The function F is the unique function defined by F ` A = B, provided that the well-definedness condition holds. (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 -> ( Fun F <-> A. y e. X A. z e. X ( [_ y / x ]_ A = [_ z / x ]_ A -> [_ y / x ]_ B = [_ z / x ]_ B ) ) )
Theorem	fliftf 5995*	The domain and range of the function 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 -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )
Theorem	fliftval 5996*	The value of the function 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 )   &    |- ( x = Y -> A = C )   &    |- ( x = Y -> B = D )   &    |- ( ph -> Fun F )   =>    |- ( ( ph /\ Y e. X ) -> ( F ` C ) = D )
Theorem	isoeq1 5997	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
|- ( H = G -> ( H Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) )
Theorem	isoeq2 5998	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
|- ( R = T -> ( H Isom R , S ( A , B ) <-> H Isom T , S ( A , B ) ) )
Theorem	isoeq3 5999	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
|- ( S = T -> ( H Isom R , S ( A , B ) <-> H Isom R , T ( A , B ) ) )
Theorem	isoeq4 6000	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
|- ( A = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) )