P. 66 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6501-6600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elabrex 6501*	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 6502*	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 6503*	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 6504*	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 6505*	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 6506*	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 6507*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 6506 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 6508*	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 6509*	Membership in the union of the range of a function. See elunirnALT 6510 for a shorter proof which uses ax-pow 4843. (Contributed by NM, 24-Sep-2006.)
|- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )
Theorem	elunirnALT 6510*	Alternate proof of elunirn 6509. It is shorter but requires ax-pow 4843 (through eluniima 6508, funiunfv 6506, ndmfv 6218). (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	fnunirn 6511*	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 6512*	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	dff13f 6513*	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	f1veqaeq 6514	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	f1cofveqaeq 6515	If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.)
|- ( ( ( F : B -1-1-> C /\ G : A -1-1-> B ) /\ ( X e. A /\ Y e. A ) ) -> ( ( F ` ( G ` X ) ) = ( F ` ( G ` Y ) ) -> X = Y ) )
Theorem	f1cofveqaeqALT 6516	Alternate proof of f1cofveqaeq 6515, 1 essential step shorter, but having more bytes (305 vs. 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( ( F : B -1-1-> C /\ G : A -1-1-> B ) /\ ( X e. A /\ Y e. A ) ) -> ( ( F ` ( G ` X ) ) = ( F ` ( G ` Y ) ) -> X = Y ) )
Theorem	2f1fvneq 6517	If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( ( ( E : D -1-1-> R /\ F : C -1-1-> D ) /\ ( A e. C /\ B e. C ) /\ A =/= B ) -> ( ( ( E ` ( F ` A ) ) = X /\ ( E ` ( F ` B ) ) = Y ) -> X =/= Y ) )
Theorem	f1mpt 6518*	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 6519	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 6520	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 6521	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 6522	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 6523	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	fpropnf1 6524	A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
|- F = { <. X , Z >. , <. Y , Z >. }   =>    |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ X =/= Y ) -> ( Fun F /\ -. Fun `' F ) )
Theorem	f1dom3fv3dif 6525	The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.)
|- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )   &    |- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )   &    |- ( ph -> F : { A , B , C } -1-1-> R )   =>    |- ( ph -> ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) )
Theorem	f1dom3el3dif 6526*	The range of a 1-1 function from a set with three different elements has (at least) three different elements. (Contributed by AV, 20-Mar-2019.)
|- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )   &    |- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )   &    |- ( ph -> F : { A , B , C } -1-1-> R )   =>    |- ( ph -> E. x e. R E. y e. R E. z e. R ( x =/= y /\ x =/= z /\ y =/= z ) )
Theorem	dff14a 6527*	A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) ) )
Theorem	dff14b 6528*	A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. ( A \ { x } ) ( F ` x ) =/= ( F ` y ) ) )
Theorem	f12dfv 6529	A one-to-one function with a domain with at least two different elements in terms of function values. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
|- A = { X , Y }   =>    |- ( ( ( X e. U /\ Y e. V ) /\ X =/= Y ) -> ( F : A -1-1-> B <-> ( F : A --> B /\ ( F ` X ) =/= ( F ` Y ) ) ) )
Theorem	f13dfv 6530	A one-to-one function with a domain with at least three different elements in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
|- A = { X , Y , Z }   =>    |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( F : A -1-1-> B <-> ( F : A --> B /\ ( ( F ` X ) =/= ( F ` Y ) /\ ( F ` X ) =/= ( F ` Z ) /\ ( F ` Y ) =/= ( F ` Z ) ) ) ) )
Theorem	dff1o6 6531*	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 6532	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 6533	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 6534	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 6535	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	nvof1o 6536	An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
|- ( ( F Fn A /\ `' F = F ) -> F : A -1-1-onto-> A )
Theorem	nvocnv 6537*	The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.)
|- ( ( F : A --> A /\ A. x e. A ( F ` ( F ` x ) ) = x ) -> `' F = F )
Theorem	fsnex 6538*	Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)
|- ( x = ( f ` A ) -> ( ps <-> ph ) )   =>    |- ( A e. V -> ( E. f ( f : { A } --> D /\ ph ) <-> E. x e. D ps ) )
Theorem	f1prex 6539*	Relate a one-to-one function with a pair as domain and two different variables. (Contributed by Thierry Arnoux, 12-Jul-2020.)
|- ( x = ( f ` A ) -> ( ps <-> ch ) )   &    |- ( y = ( f ` B ) -> ( ch <-> ph ) )   =>    |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( E. f ( f : { A , B } -1-1-> D /\ ph ) <-> E. x e. D E. y e. D ( x =/= y /\ ps ) ) )
Theorem	f1ocnvdm 6540	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 6541	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 6542	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 6543	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 6544*	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 6545*	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 6546	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 6547	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	fcof1oinvd 6548	Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 6551. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
|- ( ph -> F : A -1-1-onto-> B )   &    |- ( ph -> G : B --> A )   &    |- ( ph -> ( F o. G ) = ( _I |` B ) )   =>    |- ( ph -> `' F = G )
Theorem	fcof1od 6549	A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 6542 and fcofo 6543. Formerly part of proof of fcof1o 6551. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> G : B --> A )   &    |- ( ph -> ( G o. F ) = ( _I |` A ) )   &    |- ( ph -> ( F o. G ) = ( _I |` B ) )   =>    |- ( ph -> F : A -1-1-onto-> B )
Theorem	2fcoidinvd 6550	Show that a function is the inverse of a function if their compositions are the identity functions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> G : B --> A )   &    |- ( ph -> ( G o. F ) = ( _I |` A ) )   &    |- ( ph -> ( F o. G ) = ( _I |` B ) )   =>    |- ( ph -> `' F = G )
Theorem	fcof1o 6551	Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by AV, 15-Dec-2019.)
|- ( ( ( 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	2fvcoidd 6552*	Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first function. (Contributed by AV, 15-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> G : B --> A )   &    |- ( ph -> A. a e. A ( G ` ( F ` a ) ) = a )   =>    |- ( ph -> ( G o. F ) = ( _I |` A ) )
Theorem	2fvidf1od 6553*	A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> G : B --> A )   &    |- ( ph -> A. a e. A ( G ` ( F ` a ) ) = a )   &    |- ( ph -> A. b e. B ( F ` ( G ` b ) ) = b )   =>    |- ( ph -> F : A -1-1-onto-> B )
Theorem	2fvidinvd 6554*	Show that two functions are inverse to each other by applying them twice to each value of their domains. (Contributed by AV, 13-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> G : B --> A )   &    |- ( ph -> A. a e. A ( G ` ( F ` a ) ) = a )   &    |- ( ph -> A. b e. B ( F ` ( G ` b ) ) = b )   =>    |- ( ph -> `' F = G )
Theorem	foeqcnvco 6555	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 6556	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 6557	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 6558*	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 6559*	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 6560*	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 6561*	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 6562*	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 6563*	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 6564*	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 6565*	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 6566*	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 6567	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 6568	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 6569	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 6570	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 6571	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 6572	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 6573	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	isof1oidb 6574	A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.)
|- ( H : A -1-1-onto-> B <-> H Isom _I , _I ( A , B ) )
Theorem	isof1oopb 6575	A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.)
|- ( H : A -1-1-onto-> B <-> H Isom ( _V X. _V ) , ( _V X. _V ) ( A , B ) )
Theorem	isorel 6576	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	soisores 6577*	Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)
|- ( ( ( R Or B /\ S Or C ) /\ ( F : B --> C /\ A C_ B ) ) -> ( ( 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	soisoi 6578*	Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)
|- ( ( ( R Or A /\ S Po B ) /\ ( H : A -onto-> B /\ A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) ) ) -> H Isom R , S ( A , B ) )
Theorem	isoid 6579	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 6580	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 6581	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
|- ( H Isom R , S ( A , B ) <-> H Isom `' R , `' S ( A , B ) )
Theorem	isocnv3 6582	Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- C = ( ( A X. A ) \ R )   &    |- D = ( ( B X. B ) \ S )   =>    |- ( H Isom R , S ( A , B ) <-> H Isom C , D ( A , B ) )
Theorem	isores2 6583	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 6584	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 6585	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 6586	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	isomin 6587	Isomorphisms preserve minimal elements. Note that ( `' R " { D } ) is Takeuti and Zaring's idiom for the initial segment { x | x R D }. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
|- ( ( H Isom R , S ( A , B ) /\ ( C C_ A /\ D e. A ) ) -> ( ( C i^i ( `' R " { D } ) ) = (/) <-> ( ( H " C ) i^i ( `' S " { ( H ` D ) } ) ) = (/) ) )
Theorem	isoini 6588	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 6589	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	isofrlem 6590*	Lemma for isofr 6592. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
|- ( ph -> H Isom R , S ( A , B ) )   &    |- ( ph -> ( H " x ) e. _V )   =>    |- ( ph -> ( S Fr B -> R Fr A ) )
Theorem	isoselem 6591*	Lemma for isose 6593. (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	isofr 6592	An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( R Fr A <-> S Fr B ) )
Theorem	isose 6593	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	isofr2 6594	A weak form of isofr 6592 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
|- ( ( H Isom R , S ( A , B ) /\ B e. V ) -> ( S Fr B -> R Fr A ) )
Theorem	isopolem 6595	Lemma for isopo 6596. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
Theorem	isopo 6596	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 6597	Lemma for isoso 6598. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
Theorem	isoso 6598	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	isowe 6599	An isomorphism preserves well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( R We A <-> S We B ) )
Theorem	isowe2 6600*	A weak form of isowe 6599 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
|- ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( S We B -> R We A ) )