P. 59 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fliftfuns 5801*	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 5802*	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 5803*	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 5804	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 5805	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 5806	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 5807	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 5808	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 5809	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 5810	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 5811	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 5812*	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 5813	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 5814	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 5815	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 5816	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 5817	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 5818	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 5819	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 5820	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 5821	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 5822	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 5823*	Lemma for isose 5824. (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 5824	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 5825	Lemma for isopo 5826. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
Theorem	isopo 5826	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 5827	Lemma for isoso 5828. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
Theorem	isoso 5828	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 5829*	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 5830*	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 5831	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 1500 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 5832	Extend class notation with restricted description binder.
class ( iota_ x e. A ph )
Definition	df-riota 5833	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 5180. (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 5834*	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 5835*	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 5836*	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 5837*	Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( A e. V -> ( iota_ x e. A ps ) e. _V )
Theorem	riotav 5838	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
|- ( iota_ x e. _V ph ) = ( iota x ph )
Theorem	riotauni 5839	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 5840*	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 5841*	Deduction version of nfriota 5842. (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 5842*	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	cbvriota 5843*	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 5844*	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 5845*	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 5846	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 5847*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
Theorem	riotasbc 5848	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 5849*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2727 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 5850	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2724 analog.) (Contributed by NM, 16-Jan-2012.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )
Theorem	riota1 5851*	Property of restricted iota. Compare iota1 5194. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )
Theorem	riota1a 5852	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 5853*	A deduction version of riota2f 5854. (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 5854*	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 5855*	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	riotaprop 5856*	Properties of a restricted definite description operator. Todo (df-riota 5833 update): can some uses of riota2f 5854 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 5857*	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 5858*	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 5859*	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 5860*	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 5861*	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 5862	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 5863*	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 5864*	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 5865*	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 5866*	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 5867*	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 5868*	Lemma for acexmid 5876. (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 5869*	Lemma for acexmid 5876. (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 5870*	Lemma for acexmid 5876. (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 5871*	Lemma for acexmid 5876. (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 5872*	Lemma for acexmid 5876. 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 5220. 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 5873*	Lemma for acexmid 5876. List the cases identified in acexmidlemcase 5872 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 5874*	Lemma for acexmid 5876. This builds on acexmidlem1 5873 by noting that every element of C is inhabited. (Note that y is not quite a function in the df-fun 5220 sense because it uses ordered pairs as described in opthreg 4557 rather than df-op 3603). The set A is also found in onsucelsucexmidlem 4530. (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 5875*	Lemma for acexmid 5876. This is acexmid 5876 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 5876*	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 7207 and df-exmid 4197 syntaxes, see exmidac 7210. (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 5877	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 )
Syntax	coprab 5878	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class { <. <. x , y >. , z >. | ph }
Syntax	cmpo 5879	Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class ( x e. A , y e. B |-> C )
Definition	df-ov 5880	Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation F and its arguments A and B- will be useful for proving meaningful theorems. For example, if class F is the operation + and arguments A and B are 3 and 2 , the expression ( 3 + 2 ) can be proved to equal 5 . This definition is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets); see ovprc1 5913 and ovprc2 5914. On the other hand, we often find uses for this definition when F is a proper class. F is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5881. (Contributed by NM, 28-Feb-1995.)
|- ( A F B ) = ( F ` <. A , B >. )
Definition	df-oprab 5881*	Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally x, y, and z are distinct, although the definition doesn't strictly require it. See df-ov 5880 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpo 6012. (Contributed by NM, 12-Mar-1995.)
|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
Definition	df-mpo 5882*	Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from x , y (in A X. B) to B ( x , y )". An extension of df-mpt 4068 for two arguments. (Contributed by NM, 17-Feb-2008.)
|- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
Theorem	oveq 5883	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( F = G -> ( A F B ) = ( A G B ) )
Theorem	oveq1 5884	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( A F C ) = ( B F C ) )
Theorem	oveq2 5885	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( C F A ) = ( C F B ) )
Theorem	oveq12 5886	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
|- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) )
Theorem	oveq1i 5887	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( A F C ) = ( B F C )
Theorem	oveq2i 5888	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( C F A ) = ( C F B )
Theorem	oveq12i 5889	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- A = B   &    |- C = D   =>    |- ( A F C ) = ( B F D )
Theorem	oveqi 5890	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
|- A = B   =>    |- ( C A D ) = ( C B D )
Theorem	oveq123i 5891	Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
|- A = C   &    |- B = D   &    |- F = G   =>    |- ( A F B ) = ( C G D )
Theorem	oveq1d 5892	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A F C ) = ( B F C ) )
Theorem	oveq2d 5893	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C F A ) = ( C F B ) )
Theorem	oveqd 5894	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D ) = ( C B D ) )
Theorem	oveq12d 5895	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A F C ) = ( B F D ) )
Theorem	oveqan12d 5896	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A F C ) = ( B F D ) )
Theorem	oveqan12rd 5897	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) )
Theorem	oveq123d 5898	Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A F C ) = ( B G D ) )
Theorem	fvoveq1d 5899	Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )
Theorem	fvoveq1 5900	Equality theorem for nested function and operation value. Closed form of fvoveq1d 5899. (Contributed by AV, 23-Jul-2022.)
|- ( A = B -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )