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Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfliftval 5701* 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 )
 
Theoremisoeq1 5702 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  G  Isom  R ,  S  ( A ,  B ) ) )
 
Theoremisoeq2 5703 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  T ,  S  ( A ,  B ) ) )
 
Theoremisoeq3 5704 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( S  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  T  ( A ,  B ) ) )
 
Theoremisoeq4 5705 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( A  =  C  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  S  ( C ,  B ) ) )
 
Theoremisoeq5 5706 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  S  ( A ,  C ) ) )
 
Theoremnfiso 5707 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 )
 
Theoremisof1o 5708 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 )
 
Theoremisorel 5709 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 ) ) )
 
Theoremisoresbr 5710* 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 ) ) )
 
Theoremisoid 5711 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 )
 
Theoremisocnv 5712 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 ) )
 
Theoremisocnv2 5713 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  `' R ,  `' S ( A ,  B ) )
 
Theoremisores2 5714 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 ) )
 
Theoremisores1 5715 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 ) )
 
Theoremisores3 5716 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 ) )
 
Theoremisotr 5717 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 ) )
 
Theoremiso0 5718 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  ( (/) ,  (/) )
 
Theoremisoini 5719 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 ) }
 ) ) )
 
Theoremisoini2 5720 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 ) )
 
Theoremisoselem 5721* Lemma for isose 5722. (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 ) )
 
Theoremisose 5722 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 ) )
 
Theoremisopolem 5723 Lemma for isopo 5724. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A ) )
 
Theoremisopo 5724 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 ) )
 
Theoremisosolem 5725 Lemma for isoso 5726. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A ) )
 
Theoremisoso 5726 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 ) )
 
Theoremf1oiso 5727* 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 ) )
 
Theoremf1oiso2 5728* 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  Restricted iota (description binder)
 
Syntaxcrio 5729 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A  ph )
 
Definitiondf-riota 5730 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 5088. (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 )
 )
 
Theoremriotaeqdv 5731* 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 ) )
 
Theoremriotabidv 5732* 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 ) )
 
Theoremriotaeqbidv 5733* 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 ) )
 
Theoremriotaexg 5734* Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( A  e.  V  ->  ( iota_ x  e.  A  ps )  e.  _V )
 
Theoremriotav 5735 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
 |-  ( iota_ x  e.  _V  ph )  =  ( iota
 x ph )
 
Theoremriotauni 5736 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 } )
 
Theoremnfriota1 5737* 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 )
 
Theoremnfriotadxy 5738* Deduction version of nfriota 5739. (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 ) )
 
Theoremnfriota 5739* 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 )
 
Theoremcbvriota 5740* 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 )
 
Theoremcbvriotav 5741* 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 )
 
Theoremcsbriotag 5742* 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 )
 )
 
Theoremriotacl2 5743 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 } )
 
Theoremriotacl 5744* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
 
Theoremriotasbc 5745 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 )
 
Theoremriotabidva 5746* Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2674 analog.) (Contributed by NM, 17-Jan-2012.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A  ps )  =  ( iota_ x  e.  A  ch ) )
 
Theoremriotabiia 5747 Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2671 analog.) (Contributed by NM, 16-Jan-2012.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( iota_ x  e.  A  ph )  =  ( iota_ x  e.  A  ps )
 
Theoremriota1 5748* Property of restricted iota. Compare iota1 5102. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota_ x  e.  A  ph )  =  x ) )
 
Theoremriota1a 5749 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 ) )
 
Theoremriota2df 5750* A deduction version of riota2f 5751. (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 ) )
 
Theoremriota2f 5751* 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 ) )
 
Theoremriota2 5752* 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 ) )
 
Theoremriotaprop 5753* Properties of a restricted definite description operator. Todo (df-riota 5730 update): can some uses of riota2f 5751 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 ) )
 
Theoremriota5f 5754* 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 )
 
Theoremriota5 5755* 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 )
 
Theoremriotass2 5756* 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 ) )
 
Theoremriotass 5757* 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 )
 )
 
Theoremmoriotass 5758* 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 )
 )
 
Theoremsnriota 5759 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 ) }
 )
 
Theoremeusvobj2 5760* 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 )
 )
 
Theoremeusvobj1 5761* 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 )
 )
 
Theoremf1ofveu 5762* 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 )
 
Theoremf1ocnvfv3 5763* 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 ) )
 
Theoremriotaund 5764* 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 )  =  (/) )
 
Theoremacexmidlema 5765* Lemma for acexmid 5773. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( { (/) }  e.  A  -> 
 ph )
 
Theoremacexmidlemb 5766* Lemma for acexmid 5773. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( (/) 
 e.  B  ->  ph )
 
Theoremacexmidlemph 5767* Lemma for acexmid 5773. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( ph  ->  A  =  B )
 
Theoremacexmidlemab 5768* Lemma for acexmid 5773. (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 )
 
Theoremacexmidlemcase 5769* Lemma for acexmid 5773. 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 5125.

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 )
 )  =  { (/) } )
 ) )
 
Theoremacexmidlem1 5770* Lemma for acexmid 5773. List the cases identified in acexmidlemcase 5769 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 ) )
 
Theoremacexmidlem2 5771* Lemma for acexmid 5773. This builds on acexmidlem1 5770 by noting that every element of  C is inhabited.

(Note that  y is not quite a function in the df-fun 5125 sense because it uses ordered pairs as described in opthreg 4471 rather than df-op 3536).

The set  A is also found in onsucelsucexmidlem 4444.

(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 ) )
 
Theoremacexmidlemv 5772* Lemma for acexmid 5773.

This is acexmid 5773 with additional distinct variable constraints, 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 )
 
Theoremacexmid 5773* 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 7067 and df-exmid 4119 syntaxes, see exmidac 7070. (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 )
 
2.6.10  Operations
 
Syntaxco 5774 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 )
 
Syntaxcoprab 5775 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
 class  { <. <. x ,  y >. ,  z >.  |  ph }
 
Syntaxcmpo 5776 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 )
 
Definitiondf-ov 5777 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 5807 and ovprc2 5808. 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 5778. (Contributed by NM, 28-Feb-1995.)
 |-  ( A F B )  =  ( F ` 
 <. A ,  B >. )
 
Definitiondf-oprab 5778* 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 5777 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 5906. (Contributed by NM, 12-Mar-1995.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
 <. <. x ,  y >. ,  z >.  /\  ph ) }
 
Definitiondf-mpo 5779* 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 3991 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 ) }
 
Theoremoveq 5780 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( F  =  G  ->  ( A F B )  =  ( A G B ) )
 
Theoremoveq1 5781 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( A  =  B  ->  ( A F C )  =  ( B F C ) )
 
Theoremoveq2 5782 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( A  =  B  ->  ( C F A )  =  ( C F B ) )
 
Theoremoveq12 5783 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveq1i 5784 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  A  =  B   =>    |-  ( A F C )  =  ( B F C )
 
Theoremoveq2i 5785 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  A  =  B   =>    |-  ( C F A )  =  ( C F B )
 
Theoremoveq12i 5786 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 )
 
Theoremoveqi 5787 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
 |-  A  =  B   =>    |-  ( C A D )  =  ( C B D )
 
Theoremoveq123i 5788 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
 |-  A  =  C   &    |-  B  =  D   &    |-  F  =  G   =>    |-  ( A F B )  =  ( C G D )
 
Theoremoveq1d 5789 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A F C )  =  ( B F C ) )
 
Theoremoveq2d 5790 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C F A )  =  ( C F B ) )
 
Theoremoveqd 5791 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D )  =  ( C B D ) )
 
Theoremoveq12d 5792 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 ) )
 
Theoremoveqan12d 5793 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 ) )
 
Theoremoveqan12rd 5794 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 ) )
 
Theoremoveq123d 5795 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 ) )
 
Theoremfvoveq1d 5796 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 ) ) )
 
Theoremfvoveq1 5797 Equality theorem for nested function and operation value. Closed form of fvoveq1d 5796. (Contributed by AV, 23-Jul-2022.)
 |-  ( A  =  B  ->  ( F `  ( A O C ) )  =  ( F `  ( B O C ) ) )
 
Theoremovanraleqv 5798* 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 ) ) )
 
Theoremimbrov2fvoveq 5799 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 ) ) )
 
Theoremnfovd 5800 Deduction version of bound-variable hypothesis builder nfov 5801. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x F )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x ( A F B ) )
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