Theorem List for Intuitionistic Logic Explorer - 6701-6800 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | elpm2 6701 |
The predicate "is a partial function". (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
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Theorem | fpm 6702 |
A total function is a partial function. (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
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Theorem | mapsspm 6703 |
Set exponentiation is a subset of partial maps. (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
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Theorem | pmsspw 6704 |
Partial maps are a subset of the power set of the Cartesian product of
its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
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Theorem | mapsspw 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.)
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Theorem | fvmptmap 6706* |
Special case of fvmpt 5610 for operator theorems. (Contributed by NM,
27-Nov-2007.)
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Theorem | map0e 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.)
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Theorem | map0b 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.)
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Theorem | map0g 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.)
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Theorem | map0 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.)
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Theorem | mapsn 6711* |
The value of set exponentiation with a singleton exponent. Theorem 98
of [Suppes] p. 89. (Contributed by NM,
10-Dec-2003.)
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Theorem | mapss 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.)
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Theorem | fdiagfn 6713* |
Functionality of the diagonal map. (Contributed by Stefan O'Rear,
24-Jan-2015.)
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Theorem | fvdiagfn 6714* |
Functionality of the diagonal map. (Contributed by Stefan O'Rear,
24-Jan-2015.)
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Theorem | mapsnconst 6715 |
Every singleton map is a constant function. (Contributed by Stefan
O'Rear, 25-Mar-2015.)
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Theorem | mapsncnv 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.)
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Theorem | mapsnf1o2 6717* |
Explicit bijection between a set and its singleton functions.
(Contributed by Stefan O'Rear, 21-Mar-2015.)
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Theorem | mapsnf1o3 6718* |
Explicit bijection in the reverse of mapsnf1o2 6717. (Contributed by
Stefan O'Rear, 24-Mar-2015.)
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2.6.27 Infinite Cartesian products
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Syntax | cixp 6719 |
Extend class notation to include infinite Cartesian products.
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Definition | df-ixp 6720* |
Definition of infinite Cartesian product of [Enderton] p. 54. Enderton
uses a bold "X" with
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 represents a class
expression containing free and thus can be thought of as
   . Normally,
is not free in ,
although this is
not a requirement of the definition. (Contributed by NM,
28-Sep-2006.)
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Theorem | dfixp 6721* |
Eliminate the expression   in df-ixp 6720, under the
assumption that and are
disjoint. This way, we can say that
is bound in
  even if it
appears free in .
(Contributed by Mario Carneiro, 12-Aug-2016.)
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Theorem | ixpsnval 6722* |
The value of an infinite Cartesian product with a singleton.
(Contributed by AV, 3-Dec-2018.)
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  ![]_ ]_](_urbrack.gif)     |
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Theorem | elixp2 6723* |
Membership in an infinite Cartesian product. See df-ixp 6720 for
discussion of the notation. (Contributed by NM, 28-Sep-2006.)
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Theorem | fvixp 6724* |
Projection of a factor of an indexed Cartesian product. (Contributed by
Mario Carneiro, 11-Jun-2016.)
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Theorem | ixpfn 6725* |
A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by
Mario Carneiro, 31-May-2014.)
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Theorem | elixp 6726* |
Membership in an infinite Cartesian product. (Contributed by NM,
28-Sep-2006.)
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Theorem | elixpconst 6727* |
Membership in an infinite Cartesian product of a constant .
(Contributed by NM, 12-Apr-2008.)
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Theorem | ixpconstg 6728* |
Infinite Cartesian product of a constant . (Contributed by Mario
Carneiro, 11-Jan-2015.)
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Theorem | ixpconst 6729* |
Infinite Cartesian product of a constant . (Contributed by NM,
28-Sep-2006.)
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Theorem | ixpeq1 6730* |
Equality theorem for infinite Cartesian product. (Contributed by NM,
29-Sep-2006.)
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Theorem | ixpeq1d 6731* |
Equality theorem for infinite Cartesian product. (Contributed by Mario
Carneiro, 11-Jun-2016.)
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Theorem | ss2ixp 6732 |
Subclass theorem for infinite Cartesian product. (Contributed by NM,
29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
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Theorem | ixpeq2 6733 |
Equality theorem for infinite Cartesian product. (Contributed by NM,
29-Sep-2006.)
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Theorem | ixpeq2dva 6734* |
Equality theorem for infinite Cartesian product. (Contributed by Mario
Carneiro, 11-Jun-2016.)
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Theorem | ixpeq2dv 6735* |
Equality theorem for infinite Cartesian product. (Contributed by Mario
Carneiro, 11-Jun-2016.)
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Theorem | cbvixp 6736* |
Change bound variable in an indexed Cartesian product. (Contributed by
Jeff Madsen, 20-Jun-2011.)
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Theorem | cbvixpv 6737* |
Change bound variable in an indexed Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | nfixpxy 6738* |
Bound-variable hypothesis builder for indexed Cartesian product.
(Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon,
15-Feb-2023.)
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Theorem | nfixp1 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.)
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Theorem | ixpprc 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 ,
which is not possible for a proper class domain. (Contributed by Mario
Carneiro, 25-Jan-2015.)
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Theorem | ixpf 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.)
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Theorem | uniixp 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.)
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Theorem | ixpexgg 6743* |
The existence of an infinite Cartesian product. is normally a
free-variable parameter in . Remark in Enderton p. 54.
(Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon,
15-Feb-2023.)
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Theorem | ixpin 6744* |
The intersection of two infinite Cartesian products. (Contributed by
Mario Carneiro, 3-Feb-2015.)
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Theorem | ixpiinm 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.)
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Theorem | ixpintm 6746* |
The intersection of a collection of infinite Cartesian products.
(Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon,
15-Feb-2023.)
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Theorem | ixp0x 6747 |
An infinite Cartesian product with an empty index set. (Contributed by
NM, 21-Sep-2007.)
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Theorem | ixpssmap2g 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.)
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Theorem | ixpssmapg 6749* |
An infinite Cartesian product is a subset of set exponentiation.
(Contributed by Jeff Madsen, 19-Jun-2011.)
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Theorem | 0elixp 6750 |
Membership of the empty set in an infinite Cartesian product.
(Contributed by Steve Rodriguez, 29-Sep-2006.)
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Theorem | ixpm 6751* |
If an infinite Cartesian product of a family    is inhabited,
every    is inhabited. (Contributed by Mario Carneiro,
22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
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Theorem | ixp0 6752 |
The infinite Cartesian product of a family    with an empty
member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim
Kingdon, 16-Feb-2023.)
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Theorem | ixpssmap 6753* |
An infinite Cartesian product is a subset of set exponentiation. Remark
in [Enderton] p. 54. (Contributed by
NM, 28-Sep-2006.)
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Theorem | resixp 6754* |
Restriction of an element of an infinite Cartesian product.
(Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro,
31-May-2014.)
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Theorem | mptelixpg 6755* |
Condition for an explicit member of an indexed product. (Contributed by
Stefan O'Rear, 4-Jan-2015.)
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Theorem | elixpsn 6756* |
Membership in a class of singleton functions. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | ixpsnf1o 6757* |
A bijection between a class and single-point functions to it.
(Contributed by Stefan O'Rear, 24-Jan-2015.)
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Theorem | mapsnf1o 6758* |
A bijection between a set and single-point functions to it.
(Contributed by Stefan O'Rear, 24-Jan-2015.)
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2.6.28 Equinumerosity
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Syntax | cen 6759 |
Extend class definition to include the equinumerosity relation
("approximately equals" symbol)
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Syntax | cdom 6760 |
Extend class definition to include the dominance relation (curly
less-than-or-equal)
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Syntax | cfn 6761 |
Extend class definition to include the class of all finite sets.
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Definition | df-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.)
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Definition | df-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.)
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Definition | df-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 " ". This definition is
meaningful whether or not we
accept the Axiom of Infinity ax-inf2 15165. (Contributed by NM,
22-Aug-2008.)
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Theorem | relen 6765 |
Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
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Theorem | reldom 6766 |
Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
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Theorem | encv 6767 |
If two classes are equinumerous, both classes are sets. (Contributed by
AV, 21-Mar-2019.)
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Theorem | bren 6768* |
Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
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Theorem | brdomg 6769* |
Dominance relation. (Contributed by NM, 15-Jun-1998.)
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Theorem | brdomi 6770* |
Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
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Theorem | brdom 6771* |
Dominance relation. (Contributed by NM, 15-Jun-1998.)
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Theorem | domen 6772* |
Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.
(Contributed by NM, 15-Jun-1998.)
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Theorem | domeng 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.)
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Theorem | ctex 6774 |
A class dominated by 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.)
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Theorem | f1oen3g 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.)
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Theorem | f1oen2g 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.)
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Theorem | f1dom2g 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.)
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Theorem | f1oeng 6778 |
The domain and range of a one-to-one, onto function are equinumerous.
(Contributed by NM, 19-Jun-1998.)
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Theorem | f1domg 6779 |
The domain of a one-to-one function is dominated by its codomain.
(Contributed by NM, 4-Sep-2004.)
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Theorem | f1oen 6780 |
The domain and range of a one-to-one, onto function are equinumerous.
(Contributed by NM, 19-Jun-1998.)
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Theorem | f1dom 6781 |
The domain of a one-to-one function is dominated by its codomain.
(Contributed by NM, 19-Jun-1998.)
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Theorem | isfi 6782* |
Express " is
finite". Definition 10.29 of [TakeutiZaring] p. 91
(whose " " is a predicate instead of a class). (Contributed by
NM, 22-Aug-2008.)
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Theorem | enssdom 6783 |
Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
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Theorem | endom 6784 |
Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94.
(Contributed by NM, 28-May-1998.)
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Theorem | enrefg 6785 |
Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
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Theorem | enref 6786 |
Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
by NM, 25-Sep-2004.)
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Theorem | eqeng 6787 |
Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
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Theorem | domrefg 6788 |
Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
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Theorem | en2d 6789* |
Equinumerosity inference from an implicit one-to-one onto function.
(Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro,
12-May-2014.)
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Theorem | en3d 6790* |
Equinumerosity inference from an implicit one-to-one onto function.
(Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro,
12-May-2014.)
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Theorem | en2i 6791* |
Equinumerosity inference from an implicit one-to-one onto function.
(Contributed by NM, 4-Jan-2004.)
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Theorem | en3i 6792* |
Equinumerosity inference from an implicit one-to-one onto function.
(Contributed by NM, 19-Jul-2004.)
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Theorem | dom2lem 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.)
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Theorem | dom2d 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.)
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Theorem | dom3d 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.)
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Theorem | dom2 6796* |
A mapping (first hypothesis) that is one-to-one (second hypothesis)
implies its domain is dominated by its codomain. and can be
read    and    , as can be inferred from their
distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
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Theorem | dom3 6797* |
A mapping (first hypothesis) that is one-to-one (second hypothesis)
implies its domain is dominated by its codomain. and can be
read    and    , as can be inferred from their
distinct variable conditions. (Contributed by Mario Carneiro,
20-May-2013.)
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Theorem | idssen 6798 |
Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.)
(Revised by Mario Carneiro, 15-Nov-2014.)
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Theorem | ssdomg 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.)
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Theorem | ener 6800 |
Equinumerosity is an equivalence relation. (Contributed by NM,
19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
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