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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremerovlem 6601* Lemma for eroprf 6602. (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 ) ) ) )
 
Theoremeroprf 6602* 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 )
 
Theoremeroprf2 6603* 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 6604* 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 6605* 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 6606* 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 6607* 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 )
 
Theoremecopovsymg 6608* Assuming the operation  F is commutative, show that the relation  .~, specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
 |- 
 .~  =  { <. 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  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  =  ( y  .+  x ) )   =>    |-  ( A  .~  B  ->  B  .~  A )
 
Theoremecopovtrng 6609* 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 Jim Kingdon, 1-Sep-2019.)
 |- 
 .~  =  { <. 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  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
 
Theoremecopoverg 6610* 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 Jim Kingdon, 1-Sep-2019.)
 |- 
 .~  =  { <. 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  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |- 
 .~  Er  ( S  X.  S )
 
Theoremth3qlem1 6611* 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 6612* 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 6613* 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 6614* 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 >. ) ]  .~  )
 
Theoremoviec 6615* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.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.)
 |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  H  e.  ( S  X.  S ) )   &    |-  ( ( ( a  e.  S  /\  b  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) )  ->  K  e.  ( S  X.  S ) )   &    |-  ( ( ( c  e.  S  /\  d  e.  S )  /\  ( t  e.  S  /\  s  e.  S ) )  ->  L  e.  ( S  X.  S ) )   &    |-  .~  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 6616* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6617 instead. (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 ) )
 
Theoremecovicom 6617* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
 |-  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 >. ]  .~  )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 )  ->  D  =  H )   &    |-  ( ( ( x  e.  S  /\  y  e.  S )  /\  ( z  e.  S  /\  w  e.  S ) )  ->  G  =  J )   =>    |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( A  .+  B )  =  ( B  .+  A ) )
 
Theoremecovass 6618* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6619 will be more useful. (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 ) ) )
 
Theoremecoviass 6619* Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
 |-  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 ) )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S ) )  ->  J  =  L )   &    |-  ( ( ( x  e.  S  /\  y  e.  S )  /\  ( z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S ) )  ->  K  =  M )   =>    |-  (
 ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( ( A  .+  B )  .+  C )  =  ( A  .+  ( B  .+  C ) ) )
 
Theoremecovdi 6620* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6621 will be more helpful. (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 ) ) )
 
Theoremecovidi 6621* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
 |-  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 ) )   &    |-  (
 ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S ) )  ->  H  =  K )   &    |-  ( ( ( x  e.  S  /\  y  e.  S )  /\  ( z  e.  S  /\  w  e.  S )  /\  ( v  e.  S  /\  u  e.  S ) )  ->  J  =  L )   =>    |-  (
 ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( A  .x.  ( B  .+  C ) )  =  ( ( A 
 .x.  B )  .+  ( A  .x.  C ) ) )
 
2.6.26  The mapping operation
 
Syntaxcmap 6622 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 6623 Extend the definition of a class to include the partial mapping operation. (Read for  A  ^pm  B, "the set of all partial functions that map from  B to  A.)
 class  ^pm
 
Definitiondf-map 6624* 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 6634). 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 6625* 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 6633). 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 6624) . See mapsspm 6656 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 6626* 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 6627* 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 6628* 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 6629 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 6630 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |- 
 ^pm  Fn  ( _V  X. 
 _V )
 
Theoremreldmmap 6631 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |- 
 Rel  dom  ^m
 
Theoremmapvalg 6632* 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 6633* 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 6634* 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 6635 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 ) )
 
Theoremelmapd 6636 Deduction form of elmapg 6635. (Contributed by BJ, 11-Apr-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  ( C  e.  ( A 
 ^m  B )  <->  C : B --> A ) )
 
Theoremmapdm0 6637 The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)
 |-  ( B  e.  V  ->  ( B  ^m  (/) )  =  { (/) } )
 
Theoremelpmg 6638 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 6639 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 6640 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 6641 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 6642 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 6643 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 6644 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 )
 
Theoremelmapfn 6645 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
 |-  ( A  e.  ( B  ^m  C )  ->  A  Fn  C )
 
Theoremelmapfun 6646 A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( A  e.  ( B  ^m  C )  ->  Fun  A )
 
Theoremelmapssres 6647 A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( ( A  e.  ( B  ^m  C ) 
 /\  D  C_  C )  ->  ( A  |`  D )  e.  ( B  ^m  D ) )
 
Theoremfpmg 6648 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 6649 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 6650 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 6651 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 6652* 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 6653 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 6654 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 6655 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 6656 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 6657 Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  ^pm  B )  C_  ~P ( B  X.  A )
 
Theoremmapsspw 6658 Set exponentiation is a subset of the power set of the Cartesian 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 6659* Special case of fvmpt 5571 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 6660 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 6661 Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  =/=  (/)  ->  ( (/)  ^m  A )  =  (/) )
 
Theoremmap0g 6662 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 6663 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 6664* 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 6665 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 6666* 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 6667* 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 6668 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 6669* 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 } ) )
 
Theoremmapsnf1o2 6670* Explicit bijection between a set and its 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 : ( B 
 ^m  S ) -1-1-onto-> B
 
Theoremmapsnf1o3 6671* Explicit bijection in the reverse of mapsnf1o2 6670. (Contributed by Stefan O'Rear, 24-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   &    |-  F  =  ( y  e.  B  |->  ( S  X.  { y } ) )   =>    |-  F : B -1-1-onto-> ( B  ^m  S )
 
2.6.27  Infinite Cartesian products
 
Syntaxcixp 6672 Extend class notation to include infinite Cartesian products.
 class  X_ x  e.  A  B
 
Definitiondf-ixp 6673* Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with  x  e.  A written underneath or as a subscript, as does Stoll p. 47. Some books use a capital pi, but we will reserve that notation for products of numbers. Usually  B represents a class expression containing  x free and thus can be thought of as  B ( x ). Normally,  x is not free in  A, although this is not a requirement of the definition. (Contributed by NM, 28-Sep-2006.)
 |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B ) }
 
Theoremdfixp 6674* Eliminate the expression  { x  |  x  e.  A } in df-ixp 6673, under the assumption that  A and  x are disjoint. This way, we can say that  x is bound in  X_ x  e.  A B even if it appears free in  A. (Contributed by Mario Carneiro, 12-Aug-2016.)
 |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
 
Theoremixpsnval 6675* The value of an infinite Cartesian product with a singleton. (Contributed by AV, 3-Dec-2018.)
 |-  ( X  e.  V  -> 
 X_ x  e.  { X } B  =  {
 f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  [_ X  /  x ]_ B ) }
 )
 
Theoremelixp2 6676* Membership in an infinite Cartesian product. See df-ixp 6673 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
 |-  ( F  e.  X_ x  e.  A  B  <->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfvixp 6677* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
 |-  ( x  =  C  ->  B  =  D )   =>    |-  ( ( F  e.  X_ x  e.  A  B  /\  C  e.  A ) 
 ->  ( F `  C )  e.  D )
 
Theoremixpfn 6678* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)
 |-  ( F  e.  X_ x  e.  A  B  ->  F  Fn  A )
 
Theoremelixp 6679* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
 |-  F  e.  _V   =>    |-  ( F  e.  X_ x  e.  A  B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremelixpconst 6680* Membership in an infinite Cartesian product of a constant  B. (Contributed by NM, 12-Apr-2008.)
 |-  F  e.  _V   =>    |-  ( F  e.  X_ x  e.  A  B  <->  F : A --> B )
 
Theoremixpconstg 6681* Infinite Cartesian product of a constant  B. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  X_ x  e.  A  B  =  ( B  ^m  A ) )
 
Theoremixpconst 6682* Infinite Cartesian product of a constant  B. (Contributed by NM, 28-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  X_ x  e.  A  B  =  ( B  ^m  A )
 
Theoremixpeq1 6683* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  =  B  -> 
 X_ x  e.  A  C  =  X_ x  e.  B  C )
 
Theoremixpeq1d 6684* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  X_ x  e.  A  C  =  X_ x  e.  B  C )
 
Theoremss2ixp 6685 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
 |-  ( A. x  e.  A  B  C_  C  -> 
 X_ x  e.  A  B  C_  X_ x  e.  A  C )
 
Theoremixpeq2 6686 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
 |-  ( A. x  e.  A  B  =  C  -> 
 X_ x  e.  A  B  =  X_ x  e.  A  C )
 
Theoremixpeq2dva 6687* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  X_ x  e.  A  B  =  X_ x  e.  A  C )
 
Theoremixpeq2dv 6688* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  X_ x  e.  A  B  =  X_ x  e.  A  C )
 
Theoremcbvixp 6689* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  X_ x  e.  A  B  =  X_ y  e.  A  C
 
Theoremcbvixpv 6690* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  X_ x  e.  A  B  =  X_ y  e.  A  C
 
Theoremnfixpxy 6691* Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/_ y X_ x  e.  A  B
 
Theoremnfixp1 6692 The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x X_ x  e.  A  B
 
Theoremixpprc 6693* A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain  A, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
 |-  ( -.  A  e.  _V 
 ->  X_ x  e.  A  B  =  (/) )
 
Theoremixpf 6694* A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
 |-  ( F  e.  X_ x  e.  A  B  ->  F : A --> U_ x  e.  A  B )
 
Theoremuniixp 6695* The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |- 
 U. X_ x  e.  A  B  C_  ( A  X.  U_ x  e.  A  B )
 
Theoremixpexgg 6696* The existence of an infinite Cartesian product.  x is normally a free-variable parameter in 
B. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon, 15-Feb-2023.)
 |-  ( ( A  e.  W  /\  A. x  e.  A  B  e.  V )  ->  X_ x  e.  A  B  e.  _V )
 
Theoremixpin 6697* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  X_ x  e.  A  ( B  i^i  C )  =  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C )
 
Theoremixpiinm 6698* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
 |-  ( E. z  z  e.  B  ->  X_ x  e.  A  |^|_ y  e.  B  C  =  |^|_ y  e.  B  X_ x  e.  A  C )
 
Theoremixpintm 6699* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
 |-  ( E. z  z  e.  B  ->  X_ x  e.  A  |^| B  =  |^|_ y  e.  B  X_ x  e.  A  y )
 
Theoremixp0x 6700 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
 |-  X_ x  e.  (/)  A  =  { (/) }
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