P. 70 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	map0 6901	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 =/= (/) ) )
Theorem	mapsn 6902*	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 >. } }
Theorem	mapss 6903	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 ) )
Theorem	fdiagfn 6904*	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 ) )
Theorem	fvdiagfn 6905*	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 } ) )
Theorem	mapsnconst 6906	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 ) } ) )
Theorem	mapsncnv 6907*	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 } ) )
Theorem	mapsnf1o2 6908*	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
Theorem	mapsnf1o3 6909*	Explicit bijection in the reverse of mapsnf1o2 6908. (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 )
2.6.28  Infinite Cartesian products
Syntax	cixp 6910	Extend class notation to include infinite Cartesian products.
class X_ x e. A B
Definition	df-ixp 6911*	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 ) }
Theorem	dfixp 6912*	Eliminate the expression { x | x e. A } in df-ixp 6911, 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 ) }
Theorem	ixpsnval 6913*	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 ) } )
Theorem	elixp2 6914*	Membership in an infinite Cartesian product. See df-ixp 6911 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 ) )
Theorem	fvixp 6915*	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 )
Theorem	ixpfn 6916*	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 )
Theorem	elixp 6917*	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 ) )
Theorem	elixpconst 6918*	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 )
Theorem	ixpconstg 6919*	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 ) )
Theorem	ixpconst 6920*	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 )
Theorem	ixpeq1 6921*	Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
|- ( A = B -> X_ x e. A C = X_ x e. B C )
Theorem	ixpeq1d 6922*	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 )
Theorem	ss2ixp 6923	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 )
Theorem	ixpeq2 6924	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 )
Theorem	ixpeq2dva 6925*	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 )
Theorem	ixpeq2dv 6926*	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 )
Theorem	cbvixp 6927*	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
Theorem	cbvixpv 6928*	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
Theorem	nfixpxy 6929*	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
Theorem	nfixp1 6930	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
Theorem	ixpprc 6931*	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 = (/) )
Theorem	ixpf 6932*	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 )
Theorem	uniixp 6933*	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 )
Theorem	ixpexgg 6934*	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 )
Theorem	ixpin 6935*	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 )
Theorem	ixpiinm 6936*	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 )
Theorem	ixpintm 6937*	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 )
Theorem	ixp0x 6938	An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
|- X_ x e. (/) A = { (/) }
Theorem	ixpssmap2g 6939*	An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6940 avoids ax-coll 4209. (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 ) )
Theorem	ixpssmapg 6940*	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 ) )
Theorem	0elixp 6941	Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
|- (/) e. X_ x e. (/) A
Theorem	ixpm 6942*	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 )
Theorem	ixp0 6943	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 = (/) )
Theorem	ixpssmap 6944*	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 )
Theorem	resixp 6945*	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 )
Theorem	mptelixpg 6946*	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 ) )
Theorem	elixpsn 6947*	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 >. } ) )
Theorem	ixpsnf1o 6948*	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 )
Theorem	mapsnf1o 6949*	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 } ) )
2.6.29  Equinumerosity
Syntax	cen 6950	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ~~
Syntax	cdom 6951	Extend class definition to include the dominance relation (curly less-than-or-equal)
class ~<_
Syntax	cfn 6952	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 6953*	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 6960. (Contributed by NM, 28-Mar-1998.)
|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
Definition	df-dom 6954*	Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6964 and domen 6965. (Contributed by NM, 28-Mar-1998.)
|- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
Definition	df-fin 6955*	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 16675. (Contributed by NM, 22-Aug-2008.)
|- Fin = { x | E. y e. om x ~~ y }
Theorem	relen 6956	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~~
Theorem	reldom 6957	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~<_
Theorem	encv 6958	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
|- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
Theorem	breng 6959*	Equinumerosity relation. This variation of bren 6960 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6960. (Revised by BTernaryTau, 23-Sep-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
Theorem	bren 6960*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
Theorem	brdom2g 6961*	Dominance relation. This variation of brdomg 6962 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6962. (Revised by BTernaryTau, 29-Nov-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomg 6962*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomi 6963*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~<_ B -> E. f f : A -1-1-> B )
Theorem	brdom 6964*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. f f : A -1-1-> B )
Theorem	domen 6965*	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 ) )
Theorem	domeng 6966*	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 ) ) )
Theorem	ctex 6967	A class dominated by om is a set. See also ctfoex 7360 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 )
Theorem	f1oen4g 6968	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6973 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1dom4g 6969	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 6974 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> A ~<_ B )
Theorem	f1oen3g 6970	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6973 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 )
Theorem	f1oen2g 6971	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6973 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 )
Theorem	f1dom2g 6972	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6974 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 )
Theorem	f1oeng 6973	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 )
Theorem	f1domg 6974	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 ) )
Theorem	f1oen 6975	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 )
Theorem	f1dom 6976	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 )
Theorem	isfi 6977*	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 )
Theorem	enssdom 6978	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
|- ~~ C_ ~<_
Theorem	endom 6979	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
|- ( A ~~ B -> A ~<_ B )
Theorem	enrefg 6980	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 )
Theorem	enref 6981	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A e. _V   =>    |- A ~~ A
Theorem	eqeng 6982	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
|- ( A e. V -> ( A = B -> A ~~ B ) )
Theorem	domrefg 6983	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
|- ( A e. V -> A ~<_ A )
Theorem	en2d 6984*	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 )
Theorem	en3d 6985*	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 )
Theorem	en2i 6986*	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
Theorem	en3i 6987*	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
Theorem	dom2lem 6988*	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 )
Theorem	dom2d 6989*	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 ) )
Theorem	dom3d 6990*	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 )
Theorem	dom2 6991*	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 )
Theorem	dom3 6992*	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 )
Theorem	idssen 6993	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- _I C_ ~~
Theorem	domssr 6994	If C is a superset of B and B dominates A, then C also dominates A. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C )
Theorem	ssdomg 6995	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 ) )
Theorem	ener 6996	Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ~~ Er _V
Theorem	ensymb 6997	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B <-> B ~~ A )
Theorem	ensym 6998	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B -> B ~~ A )
Theorem	ensymi 6999	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   =>    |- B ~~ A
Theorem	ensymd 7000	Symmetry of equinumerosity. Deduction form of ensym 6998. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A ~~ B )   =>    |- ( ph -> B ~~ A )