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	f1ocnvfv2 5901	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 5902	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 5903	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 5904	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 5905	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 5906	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 5907	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 5908*	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 5909*	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 5910	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 5911	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 5912	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 5913	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 5914	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 5915*	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 5916*	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 5917*	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 5918*	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 5919*	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 5920*	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 5921*	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 5922*	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 5923*	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 5924	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 5925	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 5926	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 5927	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 5928	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 5929	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 5930	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 5931	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 5932*	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 5933	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 5934	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 5935	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 5936	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 5937	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 5938	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 5939	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 5940	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 5941	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 5942	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 5943*	Lemma for isose 5944. (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 5944	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 5945	Lemma for isopo 5946. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
Theorem	isopo 5946	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 5947	Lemma for isoso 5948. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
Theorem	isoso 5948	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 5949*	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 5950*	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 ) )
2.6.9  Cantor's Theorem
Theorem	canth 5951	No set A is equinumerous to its power set (Cantor's theorem), i.e., no function can map A onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1546 if you want the form -. E. f f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
|- A e. _V   =>    |- -. F : A -onto-> ~P A
2.6.10  Restricted iota (description binder)
Syntax	crio 5952	Extend class notation with restricted description binder.
class ( iota_ x e. A ph )
Definition	df-riota 5953	Define restricted description binder. In case there is no unique x such that ( x e. A /\ ph ) holds, it evaluates to the empty set. See also comments for df-iota 5277. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
|- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
Theorem	riotaeqdv 5954*	Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )
Theorem	riotabidv 5955*	Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
Theorem	riotaeqbidv 5956*	Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
|- ( ph -> A = B )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )
Theorem	riotaexg 5957*	Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( A e. V -> ( iota_ x e. A ps ) e. _V )
Theorem	iotaexel 5958*	Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)
|- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota x ph ) e. _V )
Theorem	riotav 5959	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
|- ( iota_ x e. _V ph ) = ( iota x ph )
Theorem	riotauni 5960	Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )
Theorem	nfriota1 5961*	The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x ( iota_ x e. A ph )
Theorem	nfriotadxy 5962*	Deduction version of nfriota 5963. (Contributed by Jim Kingdon, 12-Jan-2019.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   &    |- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x ( iota_ y e. A ps ) )
Theorem	nfriota 5963*	A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
|- F/ x ph   &    |- F/_ x A   =>    |- F/_ x ( iota_ y e. A ph )
Theorem	cbvriotavw 5964*	Change bound variable in a restricted description binder. Version of cbvriotav 5966 with a disjoint variable condition. (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )
Theorem	cbvriota 5965*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )
Theorem	cbvriotav 5966*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )
Theorem	csbriotag 5967*	Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
|- ( A e. V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
Theorem	riotacl2 5968	Membership law for "the unique element in A such that ph." (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
Theorem	riotacl 5969*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
Theorem	riotasbc 5970	Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )
Theorem	riotabidva 5971*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2787 analog.) (Contributed by NM, 17-Jan-2012.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
Theorem	riotabiia 5972	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2784 analog.) (Contributed by NM, 16-Jan-2012.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )
Theorem	riota1 5973*	Property of restricted iota. Compare iota1 5292. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )
Theorem	riota1a 5974	Property of iota. (Contributed by NM, 23-Aug-2011.)
|- ( ( x e. A /\ E! x e. A ph ) -> ( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) )
Theorem	riota2df 5975*	A deduction version of riota2f 5976. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/ x ph   &    |- ( ph -> F/_ x B )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> B e. A )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )
Theorem	riota2f 5976*	This theorem shows a condition that allows us to represent a descriptor with a class expression B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x B   &    |- F/ x ps   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
Theorem	riota2 5977*	This theorem shows a condition that allows us to represent a descriptor with a class expression B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
|- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
Theorem	riotaeqimp 5978*	If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
|- I = ( iota_ a e. V X = A )   &    |- J = ( iota_ a e. V Y = A )   &    |- ( ph -> E! a e. V X = A )   &    |- ( ph -> E! a e. V Y = A )   =>    |- ( ( ph /\ I = J ) -> X = Y )
Theorem	riotaprop 5979*	Properties of a restricted definite description operator. Todo (df-riota 5953 update): can some uses of riota2f 5976 be shortened with this? (Contributed by NM, 23-Nov-2013.)
|- F/ x ps   &    |- B = ( iota_ x e. A ph )   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph -> ( B e. A /\ ps ) )
Theorem	riota5f 5980*	A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x B )   &    |- ( ph -> B e. A )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = B )
Theorem	riota5 5981*	A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
|- ( ph -> B e. A )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = B )
Theorem	riotass2 5982*	Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
|- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) ) /\ ( E. x e. A ph /\ E! x e. B ps ) ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ps ) )
Theorem	riotass 5983*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
|- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )
Theorem	moriotass 5984*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
|- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )
Theorem	snriota 5985	A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
|- ( E! x e. A ph -> { x e. A | ph } = { ( iota_ x e. A ph ) } )
Theorem	eusvobj2 5986*	Specify the same property in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- B e. _V   =>    |- ( E! x E. y e. A x = B -> ( E. y e. A x = B <-> A. y e. A x = B ) )
Theorem	eusvobj1 5987*	Specify the same object in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- B e. _V   =>    |- ( E! x E. y e. A x = B -> ( iota x E. y e. A x = B ) = ( iota x A. y e. A x = B ) )
Theorem	f1ofveu 5988*	There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A ( F ` x ) = C )
Theorem	f1ocnvfv3 5989*	Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) = ( iota_ x e. A ( F ` x ) = C ) )
Theorem	riotaund 5990*	Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
|- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )
Theorem	acexmidlema 5991*	Lemma for acexmid 5999. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( { (/) } e. A -> ph )
Theorem	acexmidlemb 5992*	Lemma for acexmid 5999. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( (/) e. B -> ph )
Theorem	acexmidlemph 5993*	Lemma for acexmid 5999. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( ph -> A = B )
Theorem	acexmidlemab 5994*	Lemma for acexmid 5999. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( ( ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) = (/) /\ ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) = { (/) } ) -> -. ph )
Theorem	acexmidlemcase 5995*	Lemma for acexmid 5999. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle). The cases are (1) the choice function evaluated at A equals { (/) }, (2) the choice function evaluated at B equals (/), and (3) the choice function evaluated at A equals (/) and the choice function evaluated at B equals { (/) }. Because of the way we represent the choice function y, the choice function evaluated at A is ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) and the choice function evaluated at B is ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ). Other than the difference in notation these work just as ( y ` A ) and ( y ` B ) would if y were a function as defined by df-fun 5319. Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at A equals { (/) }, then { (/) } e. A and likewise for B. (Contributed by Jim Kingdon, 7-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( A. z e. C E! v e. z E. u e. y ( z e. u /\ v e. u ) -> ( { (/) } e. A \/ (/) e. B \/ ( ( iota_ v e. A E. u e. y ( A e. u /\ v e. u ) ) = (/) /\ ( iota_ v e. B E. u e. y ( B e. u /\ v e. u ) ) = { (/) } ) ) )
Theorem	acexmidlem1 5996*	Lemma for acexmid 5999. List the cases identified in acexmidlemcase 5995 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( A. z e. C E! v e. z E. u e. y ( z e. u /\ v e. u ) -> ( ph \/ -. ph ) )
Theorem	acexmidlem2 5997*	Lemma for acexmid 5999. This builds on acexmidlem1 5996 by noting that every element of C is inhabited. (Note that y is not quite a function in the df-fun 5319 sense because it uses ordered pairs as described in opthreg 4647 rather than df-op 3675). The set A is also found in onsucelsucexmidlem 4620. (Contributed by Jim Kingdon, 5-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( A. z e. C A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) -> ( ph \/ -. ph ) )
Theorem	acexmidlemv 5998*	Lemma for acexmid 5999. This is acexmid 5999 with additional disjoint variable conditions, most notably between ph and x. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )   =>    |- ( ph \/ -. ph )
Theorem	acexmid 5999*	The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483. The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function y provides a value when z is inhabited (as opposed to nonempty as in some statements of the axiom of choice). Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). For this theorem stated using the df-ac 7384 and df-exmid 4278 syntaxes, see exmidac 7387. (Contributed by Jim Kingdon, 4-Aug-2019.)
|- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )   =>    |- ( ph \/ -. ph )
2.6.11  Operations
Syntax	co 6000	Extend class notation to include the value of an operation F (such as + ) for two arguments A and B. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous.
class ( A F B )