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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmapsspm 6801 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 6802 Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  ^pm  B )  C_  ~P ( B  X.  A )
 
Theoremmapsspw 6803 Set exponentiation is a subset of the power set of the cross product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ^m  B )  C_  ~P ( B  X.  A )
 
Theoremfvmptmap 6804* Special case of fvmpt 5602 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 6805 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 6806 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 6807 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 6808 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 6809* 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 6810 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 6811* 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 6812* 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 6813 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 6814* 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 6815* 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 6816* Explicit bijection in the reverse of mapsnf1o2 6815. (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 )
 
2.4.29  Infinite Cartesian products
 
Syntaxcixp 6817 Extend class notation to include infinite Cartesian products.
 class  X_ x  e.  A  B
 
Definitiondf-ixp 6818* 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 6819* Eliminate the expression  { x  |  x  e.  A } in df-ixp 6818, 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 6820* Membership in an infinite Cartesian product. See df-ixp 6818 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 6821* 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 6822* 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 6823* 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 6824* 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 6825* 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 6826* 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 6827* 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 6828* 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 6829 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 6830 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 6831* 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 6832* 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 6833* 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 6834* 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 6835 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 6836 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 6837* 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 6838* 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 6839* 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 6840* 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 6841* 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 6842* 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 6843* 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 6844 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
 |-  X_ x  e.  (/)  A  =  { (/) }
 
Theoremixpssmap2g 6845* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6846 avoids ax-rep 4131. (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 6846* 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 6847 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
 |-  (/)  e.  X_ x  e.  (/)  A
 
Theoremixpn0 6848 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 8110. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
 
Theoremixp0 6849 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 8110. (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 6850* 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 6851* 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 6852* 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 6853* 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 6854* 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 6855* 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 6856* 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 6857* 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 6858* 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 6859* 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 )
 )
 
2.4.30  Equinumerosity
 
Syntaxcen 6860 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
 class  ~~
 
Syntaxcdom 6861 Extend class definition to include the dominance relation (curly less-than-or-equal)
 class  ~<_
 
Syntaxcsdm 6862 Extend class definition to include the strict dominance relation (curly less-than)
 class  ~<
 
Syntaxcfn 6863 Extend class definition to include the class of all finite sets.
 class  Fin
 
Definitiondf-en 6864* 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 6871. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
 
Definitiondf-dom 6865* Define the dominance relation. For an alternate definition see dfdom2 6887. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6874 and domen 6875. (Contributed by NM, 28-Mar-1998.)
 |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
 
Definitiondf-sdom 6866 Define the strict dominance relation. Alternate possible definitions are derived as brsdom 6884 and brsdom2 6985. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~<  =  (  ~<_  \  ~~  )
 
Definitiondf-fin 6867* 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 7342. If we accept Infinity, we can also express  A  e.  Fin by  A 
~<  om (theorem isfinite 7353.) (Contributed by NM, 22-Aug-2008.)
 |- 
 Fin  =  { x  |  E. y  e.  om  x  ~~  y }
 
Theoremrelen 6868 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~~
 
Theoremreldom 6869 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~<_
 
Theoremrelsdom 6870 Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
 |- 
 Rel  ~<
 
Theorembren 6871* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B )
 
Theorembrdomg 6872* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
 
Theorembrdomi 6873* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~<_  B  ->  E. f  f : A -1-1-> B )
 
Theorembrdom 6874* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
 
Theoremdomen 6875* 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 6876* 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 6877 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6880 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 6878 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6880 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 6879 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6881 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 6880 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 6881 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 6882 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 6883 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 6884 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 6885* 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 6886 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~~  C_  ~<_
 
Theoremdfdom2 6887 Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
 |-  ~<_  =  (  ~<  u.  ~~  )
 
Theoremendom 6888 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
 |-  ( A  ~~  B  ->  A  ~<_  B )
 
Theoremsdomdom 6889 Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<  B  ->  A  ~<_  B )
 
Theoremsdomnen 6890 Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<  B  ->  -.  A  ~~  B )
 
Theorembrdom2 6891 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 6892 Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  ~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
 
Theoremenrefg 6893 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 6894 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  e.  _V   =>    |-  A  ~~  A
 
Theoremeqeng 6895 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
 |-  ( A  e.  V  ->  ( A  =  B  ->  A  ~~  B ) )
 
Theoremdomrefg 6896 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
 |-  ( A  e.  V  ->  A  ~<_  A )
 
Theoremen2d 6897* 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 6898* 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 6899* 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 6900* 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
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