P. 61 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isoeq5 6001	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 6002	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 6003	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 6004	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 6005*	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 6006	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 6007	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 6008	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 6009	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 6010	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 6011	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 6012	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 6013	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 6014	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 6015	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 6016*	Lemma for isose 6017. (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 6017	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 6018	Lemma for isopo 6019. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
Theorem	isopo 6019	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 6020	Lemma for isoso 6021. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
Theorem	isoso 6021	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 6022*	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 6023*	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 ) )
Theorem	fdmrn 6024	A different way to write F is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
|- ( Fun F <-> F : dom F --> ran F )
Theorem	rinvf1o 6025	Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
|- Fun F   &    |- `' F = F   &    |- ( F " A ) C_ B   &    |- ( F " B ) C_ A   &    |- A C_ dom F   &    |- B C_ dom F   =>    |- ( F |` A ) : A -1-1-onto-> B
2.6.9  Cantor's Theorem
Theorem	canth 6026	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 1553 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 6027	Extend class notation with restricted description binder.
class ( iota_ x e. A ph )
Definition	df-riota 6028	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 5332. (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 6029*	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 6030*	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 6031*	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 6032*	Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( A e. V -> ( iota_ x e. A ps ) e. _V )
Theorem	iotaexel 6033*	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 6034	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
|- ( iota_ x e. _V ph ) = ( iota x ph )
Theorem	riotauni 6035	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 6036*	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 6037*	Deduction version of nfriota 6038. (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 6038*	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 6039*	Change bound variable in a restricted description binder. Version of cbvriotav 6041 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 6040*	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 6041*	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 6042*	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 6043	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 6044*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
Theorem	riotasbc 6045	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 6046*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2809 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 6047	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2807 analog.) (Contributed by NM, 16-Jan-2012.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )
Theorem	riota1 6048*	Property of restricted iota. Compare iota1 5347. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )
Theorem	riota1a 6049	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 6050*	A deduction version of riota2f 6051. (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 6051*	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 6052*	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 6053*	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 6054*	Properties of a restricted definite description operator. Todo (df-riota 6028 update): can some uses of riota2f 6051 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 6055*	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 6056*	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 6057*	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 6058*	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 6059*	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 6060	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 6061*	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 6062*	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 6063*	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 6064*	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 6065*	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 6066*	Lemma for acexmid 6074. (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 6067*	Lemma for acexmid 6074. (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 6068*	Lemma for acexmid 6074. (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 6069*	Lemma for acexmid 6074. (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 6070*	Lemma for acexmid 6074. 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 5374. 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 6071*	Lemma for acexmid 6074. List the cases identified in acexmidlemcase 6070 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 6072*	Lemma for acexmid 6074. This builds on acexmidlem1 6071 by noting that every element of C is inhabited. (Note that y is not quite a function in the df-fun 5374 sense because it uses ordered pairs as described in opthreg 4698 rather than df-op 3714). The set A is also found in onsucelsucexmidlem 4671. (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 6073*	Lemma for acexmid 6074. This is acexmid 6074 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 6074*	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 7552 and df-exmid 4327 syntaxes, see exmidac 7555. (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 6075	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 6076	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class { <. <. x , y >. , z >. | ph }
Syntax	cmpo 6077	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 6078	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 6112 and ovprc2 6113. 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 6079. (Contributed by NM, 28-Feb-1995.)
|- ( A F B ) = ( F ` <. A , B >. )
Definition	df-oprab 6079*	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 6078 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 6214. (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 6080*	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 4189 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 6081	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( F = G -> ( A F B ) = ( A G B ) )
Theorem	oveq1 6082	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( A F C ) = ( B F C ) )
Theorem	oveq2 6083	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( C F A ) = ( C F B ) )
Theorem	oveq12 6084	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
|- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) )
Theorem	oveq1i 6085	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( A F C ) = ( B F C )
Theorem	oveq2i 6086	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( C F A ) = ( C F B )
Theorem	oveq12i 6087	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 6088	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
|- A = B   =>    |- ( C A D ) = ( C B D )
Theorem	oveq123i 6089	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 6090	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A F C ) = ( B F C ) )
Theorem	oveq2d 6091	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C F A ) = ( C F B ) )
Theorem	oveqd 6092	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D ) = ( C B D ) )
Theorem	oveq12d 6093	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 6094	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 6095	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 6096	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 6097	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 6098	Equality theorem for nested function and operation value. Closed form of fvoveq1d 6097. (Contributed by AV, 23-Jul-2022.)
|- ( A = B -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )
Theorem	ovanraleqv 6099*	Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
|- ( B = X -> ( ph <-> ps ) )   =>    |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C ) <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) )
Theorem	imbrov2fvoveq 6100	Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
|- ( X = Y -> ( ph <-> ps ) )   =>    |- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A ) <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) )