HomeHome Metamath Proof Explorer
Theorem List (p. 67 of 311)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-31058)
 

Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremecelqsi 6601 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 6602 "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 6603 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 6604 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
 |-  A  e.  _V   =>    |-  ( A /. R )  e.  _V
 
Theoremuniqs 6605 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
 |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A ) )
 
Theoremqsss 6606 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 6607 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 6608 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 6609 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
 |-  R  e.  _V   =>    |-  [ A ] R  =  U. ( { A } /. R )
 
Theoremecid 6610 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 6611 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 6612* 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 6613* 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 6614 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
 
Theoremecelqsdm 6615 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 6616 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 6617* 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 6618* 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 6619 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 6620 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 6621 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 6622 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 6623* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
 |-  ( R  Er  X  -> Disj  x  e.  ( A /. R ) x )
 
Theoremqsel 6624 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 6625*  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 6626*  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 6627* 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 6628* 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 6629* 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 6630* 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 6631* 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 6632* 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 6633* 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 6634* 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 6635* 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 6636* 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 6637* 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 6638 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 6639* Lemma for erov 6641 and eroprf 6642. (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 6640* Lemma for erov 6641 and eroprf 6642. (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 6641* 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 6642* 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 6643* 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 6644* 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 6645* 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 6646* 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 6647* 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 6648* 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 6649* 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 6650* 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 6651* 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 6652* 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 6653* 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 6654* 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 6655* 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 6656* 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 6657* 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 ) ) )
 
2.4.26  The mapping operation
 
Syntaxcmap 6658 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 6659 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 6660* 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 6670). 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 6661* 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 6669). 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 6660) . See mapsspm 6687 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 6662* 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 6663* 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 6664* 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 6665 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 6666 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |- 
 ^pm  Fn  ( _V  X. 
 _V )
 
Theoremreldmmap 6667 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |- 
 Rel  dom  ^m
 
Theoremmapvalg 6668* 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 6669* 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 6670* 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 6671 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 6672 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 6673 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 6674 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 6675 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 6676 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 6677 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 6678 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 6679 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 6680 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 6681 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 6682 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 6683* 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 6684 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 6685 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 6686 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 ) )
 
Theoremmapsspm 6687 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  ( A  ^m  B )  C_  ( A  ^pm  B )
 
Theorempmsspw 6688 Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  ^pm  B )  C_  ~P ( B  X.  A )
 
Theoremmapsspw 6689 Set exponentiation is a subset of the power set of the cross product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ^m  B )  C_  ~P ( B  X.  A )
 
Theoremfvmptmap 6690* Special case of fvmpt 5454 for operator theorems. (Contributed by NM, 27-Nov-2007.)
 |-  C  e.  _V   &    |-  D  e.  _V   &    |-  R  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  ( R  ^m  D )  |->  B )   =>    |-  ( A : D --> R  ->  ( F `  A )  =  C )
 
Theoremmap0e 6691 Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
 
Theoremmap0b 6692 Set exponentiation with an empty base is the empty set, provided the exponent is non-empty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  =/=  (/)  ->  ( (/)  ^m  A )  =  (/) )
 
Theoremmap0g 6693 Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A 
 ^m  B )  =  (/) 
 <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
 
Theoremmap0 6694 Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) )
 
Theoremmapsn 6695* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  { B } )  =  {
 f  |  E. y  e.  A  f  =  { <. B ,  y >. } }
 
Theoremmapss 6696 Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( B  e.  V  /\  A  C_  B )  ->  ( A  ^m  C )  C_  ( B 
 ^m  C ) )
 
Theoremfdiagfn 6697* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  B  |->  ( I  X.  { x }
 ) )   =>    |-  ( ( B  e.  V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
 
Theoremfvdiagfn 6698* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  B  |->  ( I  X.  { x }
 ) )   =>    |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `
  X )  =  ( I  X.  { X } ) )
 
Theoremmapsnconst 6699 Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   =>    |-  ( F  e.  ( B  ^m  S )  ->  F  =  ( S  X.  { ( F `  X ) } )
 )
 
Theoremmapsncnv 6700* Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   &    |-  F  =  ( x  e.  ( B 
 ^m  S )  |->  ( x `  X ) )   =>    |-  `' F  =  (
 y  e.  B  |->  ( S  X.  { y } ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31058
  Copyright terms: Public domain < Previous  Next >