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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremomopth 6901 An ordered pair theorem for finite integers. Analagous to nn0opthi 11563. (Contributed by Scott Fenton, 1-May-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( C  e.  om  /\  D  e.  om ) )  ->  ( ( ( ( A  +o  B )  .o  ( A  +o  B ) )  +o  B )  =  (
 ( ( C  +o  D )  .o  ( C  +o  D ) )  +o  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
2.4.28  Equivalence relations and classes
 
Syntaxwer 6902 Extend the definition of a wff to include the equivalence predicate.
 wff  R  Er  A
 
Syntaxcec 6903 Extend the definition of a class to include equivalence class.
 class  [ A ] R
 
Syntaxcqs 6904 Extend the definition of a class to include quotient set.
 class  ( A /. R )
 
Definitiondf-er 6905 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6906 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6925, ersymb 6919, and ertr 6920. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
 C_  R ) )
 
Theoremdfer2 6906* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  A. x A. y A. z
 ( ( x R y  ->  y R x )  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
 
Definitiondf-ec 6907 Define the  R-coset of  A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of  A modulo  R when  R is an equivalence relation (i.e. when  Er  R; see dfer2 6906). In this case,  A is a representative (member) of the equivalence class  [ A ] R, which contains all sets that are equivalent to  A. Definition of [Enderton] p. 57 uses the notation  [ A ] (subscript)  R, although we simply follow the brackets by  R since we don't have subscripted expressions. For an alternate definition, see dfec2 6908. (Contributed by NM, 23-Jul-1995.)
 |- 
 [ A ] R  =  ( R " { A } )
 
Theoremdfec2 6908* Alternate definition of  R-coset of  A. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  [ A ] R  =  { y  |  A R y } )
 
Theoremecexg 6909 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
 |-  ( R  e.  B  ->  [ A ] R  e.  _V )
 
Theoremecexr 6910 A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  [ B ] R  ->  B  e.  _V )
 
Definitiondf-qs 6911* Define quotient set.  R is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
 |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
 
Theoremereq1 6912 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
 
Theoremereq2 6913 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
 
Theoremerrel 6914 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  Rel  R )
 
Theoremerdm 6915 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  dom  R  =  A )
 
Theoremercl 6916 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  A  e.  X )
 
Theoremersym 6917 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B R A )
 
Theoremercl2 6918 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B  e.  X )
 
Theoremersymb 6919 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  ( A R B  <->  B R A ) )
 
Theoremertr 6920 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  (
 ( A R B  /\  B R C ) 
 ->  A R C ) )
 
Theoremertrd 6921 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr2d 6922 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  C R A )
 
Theoremertr3d 6923 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B R A )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr4d 6924 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  A R C )
 
Theoremerref 6925 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  A R A )
 
Theoremercnv 6926 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  `' R  =  R )
 
Theoremerrn 6927 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ran  R  =  A )
 
Theoremerssxp 6928 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  R  C_  ( A  X.  A ) )
 
Theoremerex 6929 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V )
 )
 
Theoremerexb 6930 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
 
Theoremiserd 6931* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ( ph  /\  x R y )  ->  y R x )   &    |-  (
 ( ph  /\  ( x R y  /\  y R z ) ) 
 ->  x R z )   &    |-  ( ph  ->  ( x  e.  A  <->  x R x ) )   =>    |-  ( ph  ->  R  Er  A )
 
Theorembrdifun 6932 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B 
 <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
 
Theoremswoer 6933* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   =>    |-  ( ph  ->  R  Er  X )
 
Theoremswoord1 6934* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
 
Theoremswoord2 6935* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
 
Theoremswoso 6936* If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  (
 ( ph  /\  ( x  e.  Y  /\  y  e.  Y  /\  x R y ) )  ->  x  =  y )   =>    |-  ( ph  ->  .<  Or  Y )
 
Theoremeqerlem 6937* Lemma for eqer 6938. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
 
Theoremeqer 6938* Equivalence relation involving equality of dependent classes  A
( x ) and  B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  R  Er  _V
 
Theoremider 6939 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |- 
 _I  Er  _V
 
Theorem0er 6940 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  (/)  Er  (/)
 
Theoremeceq1 6941 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq2 6942 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
 
Theoremelecg 6943 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremelec 6944 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  [ B ] R  <->  B R A )
 
Theoremrelelec 6945 Membership in an equivalence class when  R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( Rel  R  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremecss 6946 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 6947 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 6948 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 6949 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 6950 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 6951 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 6952 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 6953 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
 |- 
 [ A ]  _I  =  { A }
 
Theoremqseq1 6954 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C ) )
 
Theoremqseq2 6955 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B ) )
 
Theoremelqsg 6956* Closed form of elqs 6957. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R ) )
 
Theoremelqs 6957* 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 6958* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R )
 
Theoremecelqsg 6959 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 6960 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 6961 "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 6962 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 6963 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
 |-  A  e.  _V   =>    |-  ( A /. R )  e.  _V
 
Theoremuniqs 6964 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
 |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A ) )
 
Theoremqsss 6965 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 6966 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 6967 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 6968 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
 |-  R  e.  _V   =>    |-  [ A ] R  =  U. ( { A } /. R )
 
Theoremecid 6969 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 6970 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 6971* 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 6972* 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 6973 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
 
Theoremecelqsdm 6974 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 6975 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 6976* 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 6977* 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 6978 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 6979 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 6980 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 6981 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 6982* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
 |-  ( R  Er  X  -> Disj  x  e.  ( A /. R ) x )
 
Theoremqsel 6983 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 )
 
Theoremuniinqs 6984 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4035. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)
 |-  R  Er  X   =>    |-  ( ( B 
 C_  ( A /. R )  /\  C  C_  ( A /. R ) )  ->  U. ( B  i^i  C )  =  ( U. B  i^i  U. C ) )
 
Theoremqliftlem 6985*  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 6986*  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 6987* 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 6988* 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 6989* 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 6990* 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 6991* 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 6992* 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 6993* 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 6994* 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 6995* 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 6996* 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 6997* 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 6998 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 6999* Lemma for erov 7001 and eroprf 7002. (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 7000* Lemma for erov 7001 and eroprf 7002. (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 ) ) ) )
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