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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelmapg 6801 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremelpmg 6802 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2g 6803 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)

Theoremelpm2r 6804 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)

Theoremelpmi 6805 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)

Theorempmfun 6806 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremelmapex 6807 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremelmapi 6808 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremfpmg 6809 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmss12g 6810 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmresg 6811 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theoremelmap 6812 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)

Theoremmapval2 6813* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Theoremelpm 6814 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2 6815 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremfpm 6816 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremmapsspm 6817 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)

Theorempmsspw 6818 Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremmapsspw 6819 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.)

Theoremfvmptmap 6820* Special case of fvmpt 5618 for operator theorems. (Contributed by NM, 27-Nov-2007.)

Theoremmap0e 6821 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.)

Theoremmap0b 6822 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.)

Theoremmap0g 6823 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.)

Theoremmap0 6824 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.)

Theoremmapsn 6825* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)

Theoremmapss 6826 Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfdiagfn 6827* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremfvdiagfn 6828* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremmapsnconst 6829 Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremmapsncnv 6830* 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.)

Theoremmapsnf1o2 6831* Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremmapsnf1o3 6832* Explicit bijection in the reverse of mapsnf1o2 6831. (Contributed by Stefan O'Rear, 24-Mar-2015.)

2.4.29  Infinite Cartesian products

Syntaxcixp 6833 Extend class notation to include infinite Cartesian products.

Definitiondf-ixp 6834* 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.)

Theoremdfixp 6835* Eliminate the expression in df-ixp 6834, 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.)

Theoremelixp2 6836* Membership in an infinite Cartesian product. See df-ixp 6834 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)

Theoremfvixp 6837* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremixpfn 6838* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremelixp 6839* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)

Theoremelixpconst 6840* Membership in an infinite Cartesian product of a constant . (Contributed by NM, 12-Apr-2008.)

Theoremixpconstg 6841* Infinite Cartesian product of a constant . (Contributed by Mario Carneiro, 11-Jan-2015.)

Theoremixpconst 6842* Infinite Cartesian product of a constant . (Contributed by NM, 28-Sep-2006.)

Theoremixpeq1 6843* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Theoremixpeq1d 6844* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremss2ixp 6845 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)

Theoremixpeq2 6846 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Theoremixpeq2dva 6847* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremixpeq2dv 6848* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremcbvixp 6849* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremcbvixpv 6850* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfixp 6851 Bound-variable hypothesis builder for indexed cross product. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremnfixp1 6852 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.)

Theoremixpprc 6853* 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.)

Theoremixpf 6854* 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.)

Theoremuniixp 6855* 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.)

Theoremixpexg 6856* 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 Mario Carneiro, 25-Jan-2015.)

Theoremixpin 6857* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremixpiin 6858* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremixpint 6859* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremixp0x 6860 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)

Theoremixpssmap2g 6861* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6862 avoids ax-rep 4147. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremixpssmapg 6862* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theorem0elixp 6863 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)

Theoremixpn0 6864 The infinite Cartesian product of a family with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8126. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremixp0 6865 The infinite Cartesian product of a family with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8126. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)

Theoremixpssmap 6866* An infinite Cartesian product is a subset of set exponentation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)

Theoremresixp 6867* Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Theoremundifixp 6868* Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)

Theoremmptelixpg 6869* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)

Theoremresixpfo 6870* 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.)

Theoremelixpsn 6871* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremixpsnf1o 6872* A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremmapsnf1o 6873* A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremboxriin 6874* 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.)

Theoremboxcutc 6875* The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)

2.4.30  Equinumerosity

Syntaxcen 6876 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)

Syntaxcdom 6877 Extend class definition to include the dominance relation (curly less-than-or-equal)

Syntaxcsdm 6878 Extend class definition to include the strict dominance relation (curly less-than)

Syntaxcfn 6879 Extend class definition to include the class of all finite sets.

Definitiondf-en 6880* 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 6887. (Contributed by NM, 28-Mar-1998.)

Definitiondf-dom 6881* Define the dominance relation. For an alternate definition see dfdom2 6903. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6890 and domen 6891. (Contributed by NM, 28-Mar-1998.)

Definitiondf-sdom 6882 Define the strict dominance relation. Alternate possible definitions are derived as brsdom 6900 and brsdom2 7001. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)

Definitiondf-fin 6883* 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 7358. If we accept Infinity, we can also express by (theorem isfinite 7369.) (Contributed by NM, 22-Aug-2008.)

Theoremrelen 6884 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)

Theoremreldom 6885 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)

Theoremrelsdom 6886 Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)

Theorembren 6887* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)

Theorembrdomg 6888* Dominance relation. (Contributed by NM, 15-Jun-1998.)

Theorembrdomi 6889* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorembrdom 6890* Dominance relation. (Contributed by NM, 15-Jun-1998.)

Theoremdomen 6891* Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)

Theoremdomeng 6892* 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.)

Theoremf1oen3g 6893 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6896 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremf1oen2g 6894 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6896 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremf1dom2g 6895 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6897 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremf1oeng 6896 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)

Theoremf1domg 6897 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)

Theoremf1oen 6898 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)

Theoremf1dom 6899 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)

Theorembrsdom 6900 Strict dominance relation, meaning " is strictly greater in size than ." Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)

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