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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoviec 6501* 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 6502* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6503 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 6503* 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 6504* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6505 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 6505* 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 6506* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6507 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 6507* 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 ) ) )
 
2.6.25  The mapping operation
 
Syntaxcmap 6508 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 6509 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 6510* 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 6520). 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 6511* 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 6519). 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 6510) . See mapsspm 6542 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 6512* 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 6513* 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 6514* 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 6515 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 6516 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |- 
 ^pm  Fn  ( _V  X. 
 _V )
 
Theoremreldmmap 6517 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |- 
 Rel  dom  ^m
 
Theoremmapvalg 6518* 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 6519* 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 6520* 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 6521 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 6522 Deduction form of elmapg 6521. (Contributed by BJ, 11-Apr-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  ( C  e.  ( A 
 ^m  B )  <->  C : B --> A ) )
 
Theoremmapdm0 6523 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 6524 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 6525 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 6526 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 6527 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 6528 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 6529 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 6530 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 6531 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
 |-  ( A  e.  ( B  ^m  C )  ->  A  Fn  C )
 
Theoremelmapfun 6532 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 6533 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 6534 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 6535 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 6536 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 6537 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 6538* 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 6539 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 6540 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 6541 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 6542 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 6543 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 6544 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 6545* Special case of fvmpt 5464 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 6546 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 6547 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 6548 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 6549 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 6550* 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 6551 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 6552* 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 6553* 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 6554 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 6555* 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 6556* 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 6557* Explicit bijection in the reverse of mapsnf1o2 6556. (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 )
 
2.6.26  Infinite Cartesian products
 
Syntaxcixp 6558 Extend class notation to include infinite Cartesian products.
 class  X_ x  e.  A  B
 
Definitiondf-ixp 6559* 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 6560* Eliminate the expression  { x  |  x  e.  A } in df-ixp 6559, 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 6561* 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 6562* Membership in an infinite Cartesian product. See df-ixp 6559 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 6563* 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 6564* 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 6565* 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 6566* 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 6567* 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 6568* 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 6569* 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 6570* 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 6571 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 6572 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 6573* 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 6574* 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 6575* 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 6576* 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 6577* 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 6578 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 6579* 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 6580* 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 6581* 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 6582* 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 6583* 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 6584* 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 6585* 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 6586 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
 |-  X_ x  e.  (/)  A  =  { (/) }
 
Theoremixpssmap2g 6587* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6588 avoids ax-coll 4011. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( U_ x  e.  A  B  e.  V  -> 
 X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A ) )
 
Theoremixpssmapg 6588* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( A. x  e.  A  B  e.  V  -> 
 X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A ) )
 
Theorem0elixp 6589 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
 |-  (/)  e.  X_ x  e.  (/)  A
 
Theoremixpm 6590* If an infinite Cartesian product of a family  B ( x ) is inhabited, every  B ( x ) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
 |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  E. z  z  e.  B )
 
Theoremixp0 6591 The infinite Cartesian product of a family  B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
 |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
 
Theoremixpssmap 6592* An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
 |-  B  e.  _V   =>    |-  X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A )
 
Theoremresixp 6593* Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  ( F  |`  B )  e.  X_ x  e.  B  C )
 
Theoremmptelixpg 6594* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
 |-  ( I  e.  V  ->  ( ( x  e.  I  |->  J )  e.  X_ x  e.  I  K 
 <-> 
 A. x  e.  I  J  e.  K )
 )
 
Theoremelixpsn 6595* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e.  V  ->  ( F  e.  X_ x  e.  { A } B  <->  E. y  e.  B  F  =  { <. A ,  y >. } ) )
 
Theoremixpsnf1o 6596* A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )   =>    |-  ( I  e.  V  ->  F : A
 -1-1-onto-> X_ y  e.  { I } A )
 
Theoremmapsnf1o 6597* A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )   =>    |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> ( A  ^m  { I } ) )
 
2.6.27  Equinumerosity
 
Syntaxcen 6598 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
 class  ~~
 
Syntaxcdom 6599 Extend class definition to include the dominance relation (curly less-than-or-equal)
 class  ~<_
 
Syntaxcfn 6600 Extend class definition to include the class of all finite sets.
 class  Fin
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