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	fconst2 5901	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 5902*	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5901. (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 5903*	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 5904*	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 5905	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 5906	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 5905. (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 5907	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 5908	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 5909	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 5906. (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 5910*	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 5911*	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 5912*	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 5913*	If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 5911. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. V )   =>    |- ( ph -> ( x e. A |-> B ) e. _V )
Theorem	mptrabex 5914*	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 5915	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 5916	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 5917*	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 5918	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 5919	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 5920	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 5921	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 5922	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	resfvresima 5923	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 5924*	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 5595). (Contributed by BJ, 6-Jul-2022.)
|- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) )
Theorem	foelrn 5925*	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 5926	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 5927*	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 5928*	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 5929*	TODO: This is the same as issref 5145 (which has a much longer proof). Should we replace issref 5145 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 5930*	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 5931*	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 5932*	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 5933*	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 5934*	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 5935*	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 5936*	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 5935. (Contributed by Jim Kingdon, 10-Jan-2019.)
|- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )
Theorem	funiunfvdmf 5937*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5936 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 5938*	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 5939*	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 5940*	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 5941*	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 5942	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 5943*	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 5944*	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 5945	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 5946	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 5947	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 5948	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 5949*	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 5950	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 5951	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 5952	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 5953	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 5954	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 5955	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 5956	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 5957	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 5958*	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 5959*	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 5960	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 5961	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 5962	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 5963	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 5964	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 5965*	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 5966*	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 5967*	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 5968*	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 5969*	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 5970*	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 5971*	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 5972*	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 5973*	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 5974	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 5975	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 5976	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 5977	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
|- ( A = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( C , B ) ) )
Theorem	isoeq5 5978	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
|- ( B = C -> ( H Isom R , S ( A , B ) <-> H Isom R , S ( A , C ) ) )
Theorem	nfiso 5979	Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- F/_ x H   &    |- F/_ x R   &    |- F/_ x S   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/ x H Isom R , S ( A , B )
Theorem	isof1o 5980	An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
|- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
Theorem	isorel 5981	An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
|- ( ( H Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) -> ( C R D <-> ( H ` C ) S ( H ` D ) ) )
Theorem	isoresbr 5982*	A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
|- ( ( F |` A ) Isom R , S ( A , ( F " A ) ) -> A. x e. A A. y e. A ( x R y -> ( F ` x ) S ( F ` y ) ) )
Theorem	isoid 5983	Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
|- ( _I |` A ) Isom R , R ( A , A )
Theorem	isocnv 5984	Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
|- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
Theorem	isocnv2 5985	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
|- ( H Isom R , S ( A , B ) <-> H Isom `' R , `' S ( A , B ) )
Theorem	isores2 5986	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
|- ( H Isom R , S ( A , B ) <-> H Isom R , ( S i^i ( B X. B ) ) ( A , B ) )
Theorem	isores1 5987	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
|- ( H Isom R , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )
Theorem	isores3 5988	Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ( H Isom R , S ( A , B ) /\ K C_ A /\ X = ( H " K ) ) -> ( H |` K ) Isom R , S ( K , X ) )
Theorem	isotr 5989	Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
|- ( ( H Isom R , S ( A , B ) /\ G Isom S , T ( B , C ) ) -> ( G o. H ) Isom R , T ( A , C ) )
Theorem	iso0 5990	The empty set is an R , S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
|- (/) Isom R , S ( (/) , (/) )
Theorem	isoini 5991	Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
|- ( ( H Isom R , S ( A , B ) /\ D e. A ) -> ( H " ( A i^i ( `' R " { D } ) ) ) = ( B i^i ( `' S " { ( H ` D ) } ) ) )
Theorem	isoini2 5992	Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
|- C = ( A i^i ( `' R " { X } ) )   &    |- D = ( B i^i ( `' S " { ( H ` X ) } ) )   =>    |- ( ( H Isom R , S ( A , B ) /\ X e. A ) -> ( H |` C ) Isom R , S ( C , D ) )
Theorem	isoselem 5993*	Lemma for isose 5994. (Contributed by Mario Carneiro, 23-Jun-2015.)
|- ( ph -> H Isom R , S ( A , B ) )   &    |- ( ph -> ( H " x ) e. _V )   =>    |- ( ph -> ( R Se A -> S Se B ) )
Theorem	isose 5994	An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
|- ( H Isom R , S ( A , B ) -> ( R Se A <-> S Se B ) )
Theorem	isopolem 5995	Lemma for isopo 5996. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
Theorem	isopo 5996	An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( R Po A <-> S Po B ) )
Theorem	isosolem 5997	Lemma for isoso 5998. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
Theorem	isoso 5998	An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( R Or A <-> S Or B ) )
Theorem	f1oiso 5999*	Any one-to-one onto function determines an isomorphism with an induced relation S. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
|- ( ( H : A -1-1-onto-> B /\ S = { <. z , w >. | E. x e. A E. y e. A ( ( z = ( H ` x ) /\ w = ( H ` y ) ) /\ x R y ) } ) -> H Isom R , S ( A , B ) )
Theorem	f1oiso2 6000*	Any one-to-one onto function determines an isomorphism with an induced relation S. (Contributed by Mario Carneiro, 9-Mar-2013.)
|- S = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( `' H ` x ) R ( `' H ` y ) ) }   =>    |- ( H : A -1-1-onto-> B -> H Isom R , S ( A , B ) )