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	fliftrel 6001*	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 6002*	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 6003*	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 6004*	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 6005*	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 6006*	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 6007*	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 6008*	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 6009*	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 6010	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 6011	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 6012	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 6013	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 6014	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 6015	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 6016	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 6017	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 6018*	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 6019	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 6020	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 6021	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 6022	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 6023	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 6024	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 6025	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 6026	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 6027	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 6028	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 6029*	Lemma for isose 6030. (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 6030	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 6031	Lemma for isopo 6032. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
Theorem	isopo 6032	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 6033	Lemma for isoso 6034. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
Theorem	isoso 6034	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 6035*	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 6036*	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 6037	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 6038	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 6039	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 6040	Extend class notation with restricted description binder.
class ( iota_ x e. A ph )
Definition	df-riota 6041	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 5340. (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 6042*	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 6043*	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 6044*	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 6045*	Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( A e. V -> ( iota_ x e. A ps ) e. _V )
Theorem	iotaexel 6046*	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 6047	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
|- ( iota_ x e. _V ph ) = ( iota x ph )
Theorem	riotauni 6048	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 6049*	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 6050*	Deduction version of nfriota 6051. (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 6051*	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 6052*	Change bound variable in a restricted description binder. Version of cbvriotav 6054 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 6053*	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 6054*	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 6055*	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 6056	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 6057*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
Theorem	riotasbc 6058	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 6059*	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 6060	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 6061*	Property of restricted iota. Compare iota1 5355. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )
Theorem	riota1a 6062	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 6063*	A deduction version of riota2f 6064. (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 6064*	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 6065*	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 6066*	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 6067*	Properties of a restricted definite description operator. Todo (df-riota 6041 update): can some uses of riota2f 6064 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 6068*	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 6069*	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 6070*	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 6071*	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 6072*	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 6073	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 6074*	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 6075*	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 6076*	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 6077*	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 6078*	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 6079*	Lemma for acexmid 6087. (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 6080*	Lemma for acexmid 6087. (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 6081*	Lemma for acexmid 6087. (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 6082*	Lemma for acexmid 6087. (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 6083*	Lemma for acexmid 6087. 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 5382. 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 6084*	Lemma for acexmid 6087. List the cases identified in acexmidlemcase 6083 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 6085*	Lemma for acexmid 6087. This builds on acexmidlem1 6084 by noting that every element of C is inhabited. (Note that y is not quite a function in the df-fun 5382 sense because it uses ordered pairs as described in opthreg 4705 rather than df-op 3718). The set A is also found in onsucelsucexmidlem 4678. (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 6086*	Lemma for acexmid 6087. This is acexmid 6087 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 6087*	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 7568 and df-exmid 4334 syntaxes, see exmidac 7571. (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 6088	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 6089	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class { <. <. x , y >. , z >. | ph }
Syntax	cmpo 6090	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 6091	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 6125 and ovprc2 6126. 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 6092. (Contributed by NM, 28-Feb-1995.)
|- ( A F B ) = ( F ` <. A , B >. )
Definition	df-oprab 6092*	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 6091 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 6227. (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 6093*	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 4195 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 6094	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( F = G -> ( A F B ) = ( A G B ) )
Theorem	oveq1 6095	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( A F C ) = ( B F C ) )
Theorem	oveq2 6096	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( C F A ) = ( C F B ) )
Theorem	oveq12 6097	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
|- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) )
Theorem	oveq1i 6098	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( A F C ) = ( B F C )
Theorem	oveq2i 6099	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( C F A ) = ( C F B )
Theorem	oveq12i 6100	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 )