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Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelpm2 6701 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 6702 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 6703 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 6704 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 6705 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 6706* Special case of fvmpt 5610 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 6707 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 6708 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 6709 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 6710 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 6711* 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 6712 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 6713* 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 6714* 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 6715 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 6716* 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 6717* 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 6718* Explicit bijection in the reverse of mapsnf1o2 6717. (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 6719 Extend class notation to include infinite Cartesian products.
 class  X_ x  e.  A  B
 
Definitiondf-ixp 6720* 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 6721* Eliminate the expression  { x  |  x  e.  A } in df-ixp 6720, 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 6722* 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 6723* Membership in an infinite Cartesian product. See df-ixp 6720 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 6724* 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 6725* 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 6726* 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 6727* 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 6728* 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 6729* 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 6730* 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 6731* 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 6732 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 6733 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 6734* 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 6735* 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 6736* 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 6737* 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 6738* 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 6739 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 6740* 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 6741* 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 6742* 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 6743* 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 6744* 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 6745* 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 6746* 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 6747 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
 |-  X_ x  e.  (/)  A  =  { (/) }
 
Theoremixpssmap2g 6748* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6749 avoids ax-coll 4133. (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 6749* 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 6750 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
 |-  (/)  e.  X_ x  e.  (/)  A
 
Theoremixpm 6751* 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 6752 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 6753* 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 6754* 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 6755* 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 6756* 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 6757* 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 6758* 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 } ) )
 
2.6.28  Equinumerosity
 
Syntaxcen 6759 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
 class  ~~
 
Syntaxcdom 6760 Extend class definition to include the dominance relation (curly less-than-or-equal)
 class  ~<_
 
Syntaxcfn 6761 Extend class definition to include the class of all finite sets.
 class  Fin
 
Definitiondf-en 6762* Define the equinumerosity relation. Definition of [Enderton] p. 129. We define  ~~ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 6768. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
 
Definitiondf-dom 6763* Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6771 and domen 6772. (Contributed by NM, 28-Mar-1998.)
 |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
 
Definitiondf-fin 6764* Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our " a  e.  Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 15165. (Contributed by NM, 22-Aug-2008.)
 |- 
 Fin  =  { x  |  E. y  e.  om  x  ~~  y }
 
Theoremrelen 6765 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~~
 
Theoremreldom 6766 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~<_
 
Theoremencv 6767 If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
 |-  ( A  ~~  B  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theorembren 6768* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B )
 
Theorembrdomg 6769* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
 
Theorembrdomi 6770* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~<_  B  ->  E. f  f : A -1-1-> B )
 
Theorembrdom 6771* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
 
Theoremdomen 6772* Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. x ( A 
 ~~  x  /\  x  C_  B ) )
 
Theoremdomeng 6773* Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x ( A 
 ~~  x  /\  x  C_  B ) ) )
 
Theoremctex 6774 A class dominated by  om is a set. See also ctfoex 7142 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
 |-  ( A  ~<_  om  ->  A  e.  _V )
 
Theoremf1oen3g 6775 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6778 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
 
Theoremf1oen2g 6776 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6778 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
 
Theoremf1dom2g 6777 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6779 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )
 
Theoremf1oeng 6778 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
 |-  ( ( A  e.  C  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
 
Theoremf1domg 6779 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
 |-  ( B  e.  C  ->  ( F : A -1-1-> B 
 ->  A  ~<_  B ) )
 
Theoremf1oen 6780 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
 |-  A  e.  _V   =>    |-  ( F : A
 -1-1-onto-> B  ->  A  ~~  B )
 
Theoremf1dom 6781 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( F : A -1-1-> B  ->  A  ~<_  B )
 
Theoremisfi 6782* Express " A is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose " Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
 |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x )
 
Theoremenssdom 6783 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~~  C_  ~<_
 
Theoremendom 6784 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
 |-  ( A  ~~  B  ->  A  ~<_  B )
 
Theoremenrefg 6785 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  ->  A  ~~  A )
 
Theoremenref 6786 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  e.  _V   =>    |-  A  ~~  A
 
Theoremeqeng 6787 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
 |-  ( A  e.  V  ->  ( A  =  B  ->  A  ~~  B ) )
 
Theoremdomrefg 6788 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
 |-  ( A  e.  V  ->  A  ~<_  A )
 
Theoremen2d 6789* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V ) )   &    |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  A  ~~  B )
 
Theoremen3d 6790* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <->  y  =  C ) ) )   =>    |-  ( ph  ->  A 
 ~~  B )
 
Theoremen2i 6791* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  e.  A  ->  C  e.  _V )   &    |-  ( y  e.  B  ->  D  e.  _V )   &    |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) )   =>    |-  A  ~~  B
 
Theoremen3i 6792* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( y  e.  B  ->  D  e.  A )   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  A  ~~  B
 
Theoremdom2lem 6793* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
 
Theoremdom2d 6794* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   =>    |-  ( ph  ->  ( B  e.  R  ->  A  ~<_  B ) )
 
Theoremdom3d 6795* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  A  ~<_  B )
 
Theoremdom2 6796* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
 |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y
 ) )   =>    |-  ( B  e.  V  ->  A  ~<_  B )
 
Theoremdom3 6797* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
 |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y
 ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
 
Theoremidssen 6798 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 _I  C_  ~~
 
Theoremssdomg 6799 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( B  e.  V  ->  ( A  C_  B  ->  A  ~<_  B ) )
 
Theoremener 6800 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 ~~  Er  _V
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