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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmap0b 6801 Set exponentiation with an empty base is the empty set, provided the exponent is non-empty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  =/=  (/)  ->  ( (/)  ^m  A )  =  (/) )
 
Theoremmap0g 6802 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 6803 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 6804* 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 6805 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 6806* 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 6807* 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 6808 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 6809* 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 6810* 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 6811* Explicit bijection in the reverse of mapsnf1o2 6810. (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 )
 
2.4.28  Infinite Cartesian products
 
Syntaxcixp 6812 Extend class notation to include infinite Cartesian products.
 class  X_ x  e.  A  B
 
Definitiondf-ixp 6813* 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 6814* Eliminate the expression  { x  |  x  e.  A } in df-ixp 6813, 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 ) }
 
Theoremelixp2 6815* Membership in an infinite Cartesian product. See df-ixp 6813 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 6816* 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 6817* 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 6818* 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 6819* 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 6820* 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 6821* 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 6822* 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 6823* 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 6824 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 6825 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 6826* 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 6827* 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 6828* 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 6829* 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
 
Theoremnfixp 6830 Bound-variable hypothesis builder for indexed cross product. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/_ y X_ x  e.  A  B
 
Theoremnfixp1 6831 The index variable in an indexed cross 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 6832* 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 6833* 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 6834* The union of an infinite Cartesian product is included in a cross 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 )
 
Theoremixpexg 6835* 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 Mario Carneiro, 25-Jan-2015.)
 |-  ( A. x  e.  A  B  e.  V  -> 
 X_ x  e.  A  B  e.  _V )
 
Theoremixpin 6836* 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 )
 
Theoremixpiin 6837* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)
 |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^|_ y  e.  B  C  =  |^|_ y  e.  B  X_ x  e.  A  C )
 
Theoremixpint 6838* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  ( B  =/=  (/)  ->  X_ x  e.  A  |^| B  =  |^|_ y  e.  B  X_ x  e.  A  y )
 
Theoremixp0x 6839 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
 |-  X_ x  e.  (/)  A  =  { (/) }
 
Theoremixpssmap2g 6840* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6841 avoids ax-rep 4132. (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 6841* 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 6842 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
 |-  (/)  e.  X_ x  e.  (/)  A
 
Theoremixpn0 6843 The infinite Cartesian product of a family  B ( x ) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8105. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
 
Theoremixp0 6844 The infinite Cartesian product of a family  B ( x ) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8105. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
 |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
 
Theoremixpssmap 6845* An infinite Cartesian product is a subset of set exponentation. 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 6846* 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 )
 
Theoremundifixp 6847* Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)
 |-  ( ( F  e.  X_ x  e.  B  C  /\  G  e.  X_ x  e.  ( A  \  B ) C  /\  B  C_  A )  ->  ( F  u.  G )  e.  X_ x  e.  A  C )
 
Theoremmptelixpg 6848* 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 )
 )
 
Theoremresixpfo 6849* Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  F  =  ( f  e.  X_ x  e.  A  C  |->  ( f  |`  B ) )   =>    |-  ( ( B 
 C_  A  /\  X_ x  e.  A  C  =/=  (/) )  ->  F : X_ x  e.  A  C -onto-> X_ x  e.  B  C )
 
Theoremelixpsn 6850* 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 6851* 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 6852* 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 } ) )
 
Theoremboxriin 6853* A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A. x  e.  I  A  C_  B  -> 
 X_ x  e.  I  A  =  ( X_ x  e.  I  B  i^i  |^|_
 y  e.  I  X_ x  e.  I  if ( x  =  y ,  A ,  B ) ) )
 
Theoremboxcutc 6854* The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( X  e.  A  /\  A. k  e.  A  C  C_  B )  ->  ( X_ k  e.  A  B  \  X_ k  e.  A  if ( k  =  X ,  C ,  B ) )  =  X_ k  e.  A  if ( k  =  X ,  ( B  \  C ) ,  B )
 )
 
2.4.29  Equinumerosity
 
Syntaxcen 6855 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
 class  ~~
 
Syntaxcdom 6856 Extend class definition to include the dominance relation (curly less-than-or-equal)
 class  ~<_
 
Syntaxcsdm 6857 Extend class definition to include the strict dominance relation (curly less-than)
 class  ~<
 
Syntaxcfn 6858 Extend class definition to include the class of all finite sets.
 class  Fin
 
Definitiondf-en 6859* 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 6866. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
 
Definitiondf-dom 6860* Define the dominance relation. For an alternate definition see dfdom2 6882. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6869 and domen 6870. (Contributed by NM, 28-Mar-1998.)
 |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
 
Definitiondf-sdom 6861 Define the strict dominance relation. Alternate possible definitions are derived as brsdom 6879 and brsdom2 6980. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~<  =  (  ~<_  \  ~~  )
 
Definitiondf-fin 6862* 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 7337. If we accept Infinity, we can also express  A  e.  Fin by  A 
~<  om (theorem isfinite 7348.) (Contributed by NM, 22-Aug-2008.)
 |- 
 Fin  =  { x  |  E. y  e.  om  x  ~~  y }
 
Theoremrelen 6863 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~~
 
Theoremreldom 6864 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~<_
 
Theoremrelsdom 6865 Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
 |- 
 Rel  ~<
 
Theorembren 6866* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B )
 
Theorembrdomg 6867* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
 
Theorembrdomi 6868* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~<_  B  ->  E. f  f : A -1-1-> B )
 
Theorembrdom 6869* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
 
Theoremdomen 6870* 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 6871* 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 ) ) )
 
Theoremf1oen3g 6872 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6875 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 6873 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6875 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 6874 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6876 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 6875 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 6876 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 6877 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 6878 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 )
 
Theorembrsdom 6879 Strict dominance relation, meaning " B is strictly greater in size than  A." Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  ~<  B  <->  ( A  ~<_  B  /\  -.  A  ~~  B ) )
 
Theoremisfi 6880* 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 6881 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~~  C_  ~<_
 
Theoremdfdom2 6882 Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
 |-  ~<_  =  (  ~<  u.  ~~  )
 
Theoremendom 6883 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
 |-  ( A  ~~  B  ->  A  ~<_  B )
 
Theoremsdomdom 6884 Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<  B  ->  A  ~<_  B )
 
Theoremsdomnen 6885 Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<  B  ->  -.  A  ~~  B )
 
Theorembrdom2 6886 Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
 |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B ) )
 
Theorembren2 6887 Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  ~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
 
Theoremenrefg 6888 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 6889 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  e.  _V   =>    |-  A  ~~  A
 
Theoremeqeng 6890 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
 |-  ( A  e.  V  ->  ( A  =  B  ->  A  ~~  B ) )
 
Theoremdomrefg 6891 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
 |-  ( A  e.  V  ->  A  ~<_  A )
 
Theoremen2d 6892* 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 6893* 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 6894* 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 6895* 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 6896* 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 6897* 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 6898* 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 6899* 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 6900* 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 )
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