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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremecss 6701 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  [ A ] R  C_  X )
 
Theoremecdmn0 6702 A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
 
Theoremereldm 6703 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  [ A ] R  =  [ B ] R )   =>    |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
 
Theoremerth 6704 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerth2 6705 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerthi 6706 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  [ A ] R  =  [ B ] R )
 
Theoremerdisj 6707 Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( R  Er  X  ->  ( [ A ] R  =  [ B ] R  \/  ( [ A ] R  i^i  [ B ] R )  =  (/) ) )
 
Theoremecidsn 6708 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
 |- 
 [ A ]  _I  =  { A }
 
Theoremqseq1 6709 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C ) )
 
Theoremqseq2 6710 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B ) )
 
Theoremelqsg 6711* Closed form of elqs 6712. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R ) )
 
Theoremelqs 6712* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  B  e.  _V   =>    |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R )
 
Theoremelqsi 6713* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R )
 
Theoremecelqsg 6714 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecelqsi 6715 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  R  e.  _V   =>    |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecopqsi 6716 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
 |-  R  e.  _V   &    |-  S  =  ( ( A  X.  A ) /. R )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S )
 
Theoremqsexg 6717 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A /. R )  e.  _V )
 
Theoremqsex 6718 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
 |-  A  e.  _V   =>    |-  ( A /. R )  e.  _V
 
Theoremuniqs 6719 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
 |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A ) )
 
Theoremqsss 6720 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   =>    |-  ( ph  ->  ( A /. R )  C_  ~P A )
 
Theoremuniqs2 6721 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  R  e.  V )   =>    |-  ( ph  ->  U. ( A /. R )  =  A )
 
Theoremsnec 6722 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  { [ A ] R }  =  ( { A } /. R )
 
Theoremecqs 6723 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
 |-  R  e.  _V   =>    |-  [ A ] R  =  U. ( { A } /. R )
 
Theoremecid 6724 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  [ A ] `'  _E  =  A
 
Theoremqsid 6725 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A /. `'  _E  )  =  A
 
Theoremectocld 6726* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  B )  ->  ph )   =>    |-  (
 ( ch  /\  A  e.  S )  ->  ps )
 
Theoremectocl 6727* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theoremelqsn0 6728 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
 
Theoremecelqsdm 6729 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
 |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) ) 
 ->  B  e.  A )
 
Theoremxpider 6730 A square cross product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  X.  A )  Er  A
 
Theoremiiner 6731* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  R  Er  B )  ->  |^|_
 x  e.  A  R  Er  B )
 
Theoremriiner 6732* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A. x  e.  A  R  Er  B  ->  ( ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
 
Theoremerinxp 6733 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )
 
Theoremecinxp 6734 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  ( ( ( R
 " A )  C_  A  /\  B  e.  A )  ->  [ B ] R  =  [ B ] ( R  i^i  ( A  X.  A ) ) )
 
Theoremqsinxp 6735 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ( R " A )  C_  A  ->  ( A /. R )  =  ( A /. ( R  i^i  ( A  X.  A ) ) ) )
 
Theoremqsdisj 6736 Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B  e.  ( A /. R ) )   &    |-  ( ph  ->  C  e.  ( A /. R ) )   =>    |-  ( ph  ->  ( B  =  C  \/  ( B  i^i  C )  =  (/) ) )
 
Theoremqsdisj2 6737* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
 |-  ( R  Er  X  -> Disj  x  e.  ( A /. R ) x )
 
Theoremqsel 6738 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ( R  Er  X  /\  B  e.  ( A /. R )  /\  C  e.  B )  ->  B  =  [ C ] R )
 
Theoremqliftlem 6739*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ( ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
 
Theoremqliftrel 6740*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  F  C_  ( ( X /. R )  X.  Y ) )
 
Theoremqliftel 6741* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( [ C ] R F D 
 <-> 
 E. x  e.  X  ( C R x  /\  D  =  A )
 ) )
 
Theoremqliftel1 6742* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ( ph  /\  x  e.  X )  ->  [ x ] R F A )
 
Theoremqliftfun 6743* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  y 
 ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B )
 ) )
 
Theoremqliftfund 6744* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  y 
 ->  A  =  B )   &    |-  ( ( ph  /\  x R y )  ->  A  =  B )   =>    |-  ( ph  ->  Fun  F )
 
Theoremqliftfuns 6745* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( Fun  F  <->  A. y A. z
 ( y R z 
 ->  [_ y  /  x ]_ A  =  [_ z  /  x ]_ A ) ) )
 
Theoremqliftf 6746* The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( Fun  F  <->  F : ( X
 /. R ) --> Y ) )
 
Theoremqliftval 6747* The value of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  C  ->  A  =  B )   &    |-  ( ph  ->  Fun  F )   =>    |-  ( ( ph  /\  C  e.  X )  ->  ( F `  [ C ] R )  =  B )
 
Theoremecoptocl 6748* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
 |-  S  =  ( ( B  X.  C )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( x  e.  B  /\  y  e.  C )  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theorem2ecoptocl 6749* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
 |-  S  =  ( ( C  X.  D )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( [ <. z ,  w >. ] R  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( ( x  e.  C  /\  y  e.  D )  /\  (
 z  e.  C  /\  w  e.  D )
 )  ->  ph )   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
 
Theorem3ecoptocl 6750* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
 |-  S  =  ( ( D  X.  D )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( [ <. z ,  w >. ] R  =  B  ->  ( ps  <->  ch ) )   &    |-  ( [ <. v ,  u >. ] R  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( ( x  e.  D  /\  y  e.  D )  /\  (
 z  e.  D  /\  w  e.  D )  /\  ( v  e.  D  /\  u  e.  D ) )  ->  ph )   =>    |-  (
 ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
 
Theorembrecop 6751* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( G  X.  G )   &    |-  H  =  ( ( G  X.  G ) /.  .~  )   &    |- 
 .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ] 
 .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }   &    |-  (
 ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
 )  /\  ( (
 v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G ) ) )  ->  (
 ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ] 
 .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps ) ) )   =>    |-  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
 )  ->  ( [ <. A ,  B >. ] 
 .~  .<_  [ <. C ,  D >. ]  .~  <->  ps ) )
 
Theorembrecop2 6752 Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)
 |- 
 .~  e.  _V   &    |-  dom  .~  =  ( G  X.  G )   &    |-  H  =  ( ( G  X.  G ) /.  .~  )   &    |-  R  C_  ( H  X.  H )   &    |-  .<_  C_  ( G  X.  G )   &    |-  -.  (/)  e.  G   &    |-  dom  .+  =  ( G  X.  G )   &    |-  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
 )  ->  ( [ <. A ,  B >. ] 
 .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  .<_  ( B  .+  C ) ) )   =>    |-  ( [ <. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  .<_  ( B 
 .+  C ) )
 
Theoremeroveu 6753* Lemma for erov 6755 and eroprf 6756. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   =>    |-  ( ( ph  /\  ( X  e.  J  /\  Y  e.  K )
 )  ->  E! z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T ) )
 
Theoremerovlem 6754* Lemma for erov 6755 and eroprf 6756. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   =>    |-  ( ph  ->  .+^  =  ( x  e.  J ,  y  e.  K  |->  ( iota
 z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) ) ) )
 
Theoremerov 6755* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   =>    |-  ( ( ph  /\  P  e.  A  /\  Q  e.  B )  ->  ( [ P ] R  .+^  [ Q ] S )  =  [
 ( P  .+  Q ) ] T )
 
Theoremeroprf 6756* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   &    |-  L  =  ( C
 /. T )   =>    |-  ( ph  ->  .+^  : ( J  X.  K )
 --> L )
 
Theoremerov2 6757* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  J  =  ( A
 /.  .~  )   &    |-  .+^  =  { <.
 <. x ,  y >. ,  z >.  |  E. p  e.  A  E. q  e.  A  ( ( x  =  [ p ]  .~  /\  y  =  [
 q ]  .~  )  /\  z  =  [
 ( p  .+  q
 ) ]  .~  ) }   &    |-  ( ph  ->  .~  e.  X )   &    |-  ( ph  ->  .~ 
 Er  U )   &    |-  ( ph  ->  A  C_  U )   &    |-  ( ph  ->  .+  :
 ( A  X.  A )
 --> A )   &    |-  ( ( ph  /\  ( ( r  e.  A  /\  s  e.  A )  /\  (
 t  e.  A  /\  u  e.  A )
 ) )  ->  (
 ( r  .~  s  /\  t  .~  u ) 
 ->  ( r  .+  t
 )  .~  ( s  .+  u ) ) )   =>    |-  ( ( ph  /\  P  e.  A  /\  Q  e.  A )  ->  ( [ P ]  .~  .+^  [ Q ]  .~  )  =  [
 ( P  .+  Q ) ]  .~  )
 
Theoremeroprf2 6758* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  J  =  ( A
 /.  .~  )   &    |-  .+^  =  { <.
 <. x ,  y >. ,  z >.  |  E. p  e.  A  E. q  e.  A  ( ( x  =  [ p ]  .~  /\  y  =  [
 q ]  .~  )  /\  z  =  [
 ( p  .+  q
 ) ]  .~  ) }   &    |-  ( ph  ->  .~  e.  X )   &    |-  ( ph  ->  .~ 
 Er  U )   &    |-  ( ph  ->  A  C_  U )   &    |-  ( ph  ->  .+  :
 ( A  X.  A )
 --> A )   &    |-  ( ( ph  /\  ( ( r  e.  A  /\  s  e.  A )  /\  (
 t  e.  A  /\  u  e.  A )
 ) )  ->  (
 ( r  .~  s  /\  t  .~  u ) 
 ->  ( r  .+  t
 )  .~  ( s  .+  u ) ) )   =>    |-  ( ph  ->  .+^  : ( J  X.  J ) --> J )
 
Theoremecopoveq 6759* This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation 
.~ (specified by the hypothesis) in terms of its operation  F. (Contributed by NM, 16-Aug-1995.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( <. A ,  B >.  .~  <. C ,  D >.  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
 
Theoremecopovsym 6760* Assuming the operation  F is commutative, show that the relation  .~, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( x  .+  y )  =  (
 y  .+  x )   =>    |-  ( A  .~  B  ->  B  .~  A )
 
Theoremecopovtrn 6761* Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( x  .+  y )  =  (
 y  .+  x )   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
 
Theoremecopover 6762* Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( x  .+  y )  =  (
 y  .+  x )   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |- 
 .~  Er  ( S  X.  S )
 
Theoremeceqoveq 6763* Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |- 
 .~  Er  ( S  X.  S )   &    |-  dom  .+  =  ( S  X.  S )   &    |-  -.  (/)  e.  S   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  ( <. A ,  B >.  .~  <. C ,  D >.  <->  ( A  .+  D )  =  ( B  .+  C ) ) )   =>    |-  ( ( A  e.  S  /\  C  e.  S )  ->  ( [ <. A ,  B >. ]  .~  =  [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
 
Theoremth3qlem1 6764* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |- 
 .~  Er  S   &    |-  ( ( ( y  e.  S  /\  w  e.  S )  /\  ( z  e.  S  /\  v  e.  S ) )  ->  ( ( y  .~  w  /\  z  .~  v )  ->  ( y  .+  z ) 
 .~  ( w  .+  v ) ) )   =>    |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S
 /.  .~  ) )  ->  E* x E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [
 z ]  .~  )  /\  x  =  [
 ( y  .+  z
 ) ]  .~  )
 )
 
Theoremth3qlem2 6765* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  (
 ( ( ( w  e.  S  /\  v  e.  S )  /\  ( u  e.  S  /\  t  e.  S )
 )  /\  ( (
 s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+ 
 <. s ,  f >. ) 
 .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )   =>    |-  ( ( A  e.  ( ( S  X.  S ) /.  .~  )  /\  B  e.  (
 ( S  X.  S ) /.  .~  ) ) 
 ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
 ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  )
 )
 
Theoremth3qcor 6766* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  (
 ( ( ( w  e.  S  /\  v  e.  S )  /\  ( u  e.  S  /\  t  e.  S )
 )  /\  ( (
 s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+ 
 <. s ,  f >. ) 
 .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )   &    |-  G  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  (
 ( S  X.  S ) /.  .~  ) ) 
 /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] 
 .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+ 
 <. u ,  t >. ) ]  .~  ) ) }   =>    |- 
 Fun  G
 
Theoremth3q 6767* Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  (
 ( ( ( w  e.  S  /\  v  e.  S )  /\  ( u  e.  S  /\  t  e.  S )
 )  /\  ( (
 s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+ 
 <. s ,  f >. ) 
 .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )   &    |-  G  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  (
 ( S  X.  S ) /.  .~  ) ) 
 /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] 
 .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+ 
 <. u ,  t >. ) ]  .~  ) ) }   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( [ <. A ,  B >. ] 
 .~  G [ <. C ,  D >. ]  .~  )  =  [ ( <. A ,  B >.  .+ 
 <. C ,  D >. ) ]  .~  )
 
Theoremovec 6768* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See set.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
 |-  H  e.  _V   &    |-  K  e.  _V   &    |-  L  e.  _V   &    |-  .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) ) }   &    |-  (
 ( ( z  =  a  /\  w  =  b )  /\  (
 v  =  c  /\  u  =  d )
 )  ->  ( ph  <->  ps ) )   &    |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t 
 /\  u  =  s ) )  ->  ( ph 
 <->  ch ) )   &    |-  .+  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. ) 
 /\  z  =  J ) ) }   &    |-  (
 ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h )
 )  ->  J  =  K )   &    |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t 
 /\  f  =  s ) )  ->  J  =  L )   &    |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  J  =  H )   &    |-  .+^  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  Q  /\  y  e.  Q )  /\  E. a E. b E. c E. d
 ( ( x  =  [ <. a ,  b >. ]  .~  /\  y  =  [ <. c ,  d >. ]  .~  )  /\  z  =  [ ( <. a ,  b >.  .+ 
 <. c ,  d >. ) ]  .~  ) ) }   &    |-  Q  =  ( ( S  X.  S ) /.  .~  )   &    |-  (
 ( ( ( a  e.  S  /\  b  e.  S )  /\  (
 c  e.  S  /\  d  e.  S )
 )  /\  ( (
 g  e.  S  /\  h  e.  S )  /\  ( t  e.  S  /\  s  e.  S ) ) )  ->  ( ( ps  /\  ch )  ->  K  .~  L ) )   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  ( [ <. A ,  B >. ] 
 .~  .+^  [ <. C ,  D >. ]  .~  )  =  [ H ]  .~  )
 
Theoremecovcom 6769* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
 |-  C  =  ( ( S  X.  S )
 /.  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 )  ->  ( [ <. x ,  y >. ] 
 .~  .+  [ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )   &    |-  (
 ( ( z  e.  S  /\  w  e.  S )  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( [ <. z ,  w >. ] 
 .~  .+  [ <. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )   &    |-  D  =  H   &    |-  G  =  J   =>    |-  (
 ( A  e.  C  /\  B  e.  C ) 
 ->  ( A  .+  B )  =  ( B  .+  A ) )
 
Theoremecovass 6770* Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
 |-  D  =  ( ( S  X.  S )
 /.  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 )  ->  ( [ <. x ,  y >. ] 
 .~  .+  [ <. z ,  w >. ]  .~  )  =  [ <. G ,  H >. ]  .~  )   &    |-  (
 ( ( z  e.  S  /\  w  e.  S )  /\  (
 v  e.  S  /\  u  e.  S )
 )  ->  ( [ <. z ,  w >. ] 
 .~  .+  [ <. v ,  u >. ]  .~  )  =  [ <. N ,  Q >. ]  .~  )   &    |-  (
 ( ( G  e.  S  /\  H  e.  S )  /\  ( v  e.  S  /\  u  e.  S ) )  ->  ( [ <. G ,  H >. ]  .~  .+  [ <. v ,  u >. ]  .~  )  =  [ <. J ,  K >. ]  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  ( N  e.  S  /\  Q  e.  S )
 )  ->  ( [ <. x ,  y >. ] 
 .~  .+  [ <. N ,  Q >. ]  .~  )  =  [ <. L ,  M >. ]  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 )  ->  ( G  e.  S  /\  H  e.  S ) )   &    |-  (
 ( ( z  e.  S  /\  w  e.  S )  /\  (
 v  e.  S  /\  u  e.  S )
 )  ->  ( N  e.  S  /\  Q  e.  S ) )   &    |-  J  =  L   &    |-  K  =  M   =>    |-  (
 ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( ( A  .+  B )  .+  C )  =  ( A  .+  ( B  .+  C ) ) )
 
Theoremecovdi 6771* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
 |-  D  =  ( ( S  X.  S )
 /.  .~  )   &    |-  (
 ( ( z  e.  S  /\  w  e.  S )  /\  (
 v  e.  S  /\  u  e.  S )
 )  ->  ( [ <. z ,  w >. ] 
 .~  .+  [ <. v ,  u >. ]  .~  )  =  [ <. M ,  N >. ]  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  ( M  e.  S  /\  N  e.  S )
 )  ->  ( [ <. x ,  y >. ] 
 .~  .x.  [ <. M ,  N >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 )  ->  ( [ <. x ,  y >. ] 
 .~  .x.  [ <. z ,  w >. ]  .~  )  =  [ <. W ,  X >. ]  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 v  e.  S  /\  u  e.  S )
 )  ->  ( [ <. x ,  y >. ] 
 .~  .x.  [ <. v ,  u >. ]  .~  )  =  [ <. Y ,  Z >. ]  .~  )   &    |-  (
 ( ( W  e.  S  /\  X  e.  S )  /\  ( Y  e.  S  /\  Z  e.  S ) )  ->  ( [ <. W ,  X >. ] 
 .~  .+  [ <. Y ,  Z >. ]  .~  )  =  [ <. K ,  L >. ]  .~  )   &    |-  (
 ( ( z  e.  S  /\  w  e.  S )  /\  (
 v  e.  S  /\  u  e.  S )
 )  ->  ( M  e.  S  /\  N  e.  S ) )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 )  ->  ( W  e.  S  /\  X  e.  S ) )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 v  e.  S  /\  u  e.  S )
 )  ->  ( Y  e.  S  /\  Z  e.  S ) )   &    |-  H  =  K   &    |-  J  =  L   =>    |-  (
 ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( A  .x.  ( B  .+  C ) )  =  ( ( A 
 .x.  B )  .+  ( A  .x.  C ) ) )
 
2.4.28  The mapping operation
 
Syntaxcmap 6772 Extend the definition of a class to include the mapping operation. (Read for  A  ^m  B, "the set of all functions that map from  B to  A.)
 class  ^m
 
Syntaxcpm 6773 Extend the definition of a class to include the partial mapping operation. (Read for  A  ^m  B, "the set of all partial functions that map from  B to  A.)
 class  ^pm
 
Definitiondf-map 6774* Define the mapping operation or set exponentiation. The set of all functions that map from  B to  A is written  ( A  ^m  B ) (see mapval 6784). Many authors write  A followed by  B as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g. Definition 10.42 of [TakeutiZaring] p. 95). Other authors show 
B as a prefixed superscript, which is read " A pre  B " (e.g. definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map( B,  A) for our  ( A  ^m  B ). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.)
 |- 
 ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
 )
 
Definitiondf-pm 6775* Define the partial mapping operation. A partial function from  B to  A is a function from a subset of  B to  A. The set of all partial functions from  B to  A is written  ( A  ^pm  B ) (see pmvalg 6783). A notation for this operation apparently does not appear in the literature. We use 
^pm to distinguish it from the less general set exponentiation operation  ^m (df-map 6774) . See mapsspm 6801 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
 |- 
 ^pm  =  ( x  e.  _V ,  y  e. 
 _V  |->  { f  e.  ~P ( y  X.  x )  |  Fun  f }
 )
 
Theoremmapprc 6776* When  A is a proper class, the class of all functions mapping  A to  B is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)
 |-  ( -.  A  e.  _V 
 ->  { f  |  f : A --> B }  =  (/) )
 
Theorempmex 6777* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B ) ) }  e.  _V )
 
Theoremmapex 6778* The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : A --> B }  e.  _V )
 
Theoremfnmap 6779 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 ^m  Fn  ( _V  X. 
 _V )
 
Theoremfnpm 6780 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |- 
 ^pm  Fn  ( _V  X. 
 _V )
 
Theoremreldmmap 6781 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |- 
 Rel  dom  ^m
 
Theoremmapvalg 6782* The value of set exponentiation.  ( A  ^m  B
) is the set of all functions that map from  B to  A. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^m  B )  =  {
 f  |  f : B --> A } )
 
Theorempmvalg 6783* The value of the partial mapping operation.  ( A  ^pm  B ) is the set of all partial functions that map from  B to  A. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^pm  B )  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } )
 
Theoremmapval 6784* The value of set exponentiation (inference version).  ( A  ^m  B ) is the set of all functions that map from  B to  A. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  B )  =  { f  |  f : B --> A }
 
Theoremelmapg 6785 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^m  B )  <->  C : B --> A ) )
 
Theoremelpmg 6786 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <-> 
 ( Fun  C  /\  C  C_  ( B  X.  A ) ) ) )
 
Theoremelpm2g 6787 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F  e.  ( A  ^pm  B )  <-> 
 ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
 
Theoremelpm2r 6788 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( F : C --> A  /\  C  C_  B ) ) 
 ->  F  e.  ( A 
 ^pm  B ) )
 
Theoremelpmi 6789 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
 |-  ( F  e.  ( A  ^pm  B )  ->  ( F : dom  F --> A  /\  dom  F  C_  B ) )
 
Theorempmfun 6790 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( F  e.  ( A  ^pm  B )  ->  Fun  F )
 
Theoremelmapex 6791 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  ( A  e.  ( B  ^m  C )  ->  ( B  e.  _V  /\  C  e.  _V )
 )
 
Theoremelmapi 6792 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
 
Theoremfpmg 6793 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A --> B ) 
 ->  F  e.  ( B 
 ^pm  A ) )
 
Theorempmss12g 6794 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ( ( A 
 C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W )
 )  ->  ( A  ^pm 
 B )  C_  ( C  ^pm  D ) )
 
Theorempmresg 6795 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) ) 
 ->  ( F  |`  B )  e.  ( A  ^pm  B ) )
 
Theoremelmap 6796 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F  e.  ( A  ^m  B )  <->  F : B --> A )
 
Theoremmapval2 6797* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  B )  =  ( ~P ( B  X.  A )  i^i  { f  |  f  Fn  B }
 )
 
Theoremelpm 6798 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F  e.  ( A  ^pm  B )  <->  ( Fun  F  /\  F  C_  ( B  X.  A ) ) )
 
Theoremelpm2 6799 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F 
 C_  B ) )
 
Theoremfpm 6800 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F : A --> B  ->  F  e.  ( B  ^pm  A ) )
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