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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmapsnf1o2 6701* Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremmapsnf1o3 6702* Explicit bijection in the reverse of mapsnf1o2 6701. (Contributed by Stefan O'Rear, 24-Mar-2015.)

2.4.27  Infinite Cartesian products

Syntaxcixp 6703 Extend class notation to include infinite Cartesian products.

Definitiondf-ixp 6704* 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 6705* Eliminate the expression in df-ixp 6704, 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 6706* Membership in an infinite Cartesian product. See df-ixp 6704 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)

Theoremfvixp 6707* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremixpfn 6708* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremelixp 6709* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)

Theoremelixpconst 6710* Membership in an infinite Cartesian product of a constant . (Contributed by NM, 12-Apr-2008.)

Theoremixpconstg 6711* Infinite Cartesian product of a constant . (Contributed by Mario Carneiro, 11-Jan-2015.)

Theoremixpconst 6712* Infinite Cartesian product of a constant . (Contributed by NM, 28-Sep-2006.)

Theoremixpeq1 6713* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Theoremixpeq1d 6714* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremss2ixp 6715 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)

Theoremixpeq2 6716 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Theoremixpeq2dva 6717* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremixpeq2dv 6718* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremcbvixp 6719* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremcbvixpv 6720* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfixp 6721 Bound-variable hypothesis builder for indexed cross product. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremnfixp1 6722 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 6723* 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 6724* 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 6725* 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 6726* 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 6727* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremixpiin 6728* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremixpint 6729* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremixp0x 6730 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)

Theoremixpssmap2g 6731* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6732 avoids ax-rep 4028. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremixpssmapg 6732* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theorem0elixp 6733 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)

Theoremixpn0 6734 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 7994. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremixp0 6735 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 7994. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)

Theoremixpssmap 6736* An infinite Cartesian product is a subset of set exponentation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)

Theoremresixp 6737* Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Theoremundifixp 6738* Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)

Theoremmptelixpg 6739* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)

Theoremresixpfo 6740* 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 6741* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremixpsnf1o 6742* A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremmapsnf1o 6743* A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremboxriin 6744* 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 6745* The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)

2.4.28  Equinumerosity

Syntaxcen 6746 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)

Syntaxcdom 6747 Extend class definition to include the dominance relation (curly less-than-or-equal)

Syntaxcsdm 6748 Extend class definition to include the strict dominance relation (curly less-than)

Syntaxcfn 6749 Extend class definition to include the class of all finite sets.

Definitiondf-en 6750* 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 6757. (Contributed by NM, 28-Mar-1998.)

Definitiondf-dom 6751* Define the dominance relation. For an alternate definition see dfdom2 6773. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6760 and domen 6761. (Contributed by NM, 28-Mar-1998.)

Definitiondf-sdom 6752 Define the strict dominance relation. Alternate possible definitions are derived as brsdom 6770 and brsdom2 6870. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)

Definitiondf-fin 6753* 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 7226. If we accept Infinity, we can also express by (theorem isfinite 7237.) (Contributed by NM, 22-Aug-2008.)

Theoremrelen 6754 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)

Theoremreldom 6755 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)

Theoremrelsdom 6756 Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)

Theorembren 6757* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)

Theorembrdomg 6758* Dominance relation. (Contributed by NM, 15-Jun-1998.)

Theorembrdomi 6759* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorembrdom 6760* Dominance relation. (Contributed by NM, 15-Jun-1998.)

Theoremdomen 6761* Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)

Theoremdomeng 6762* 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 6763 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6766 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremf1oen2g 6764 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6766 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremf1dom2g 6765 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6767 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremf1oeng 6766 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)

Theoremf1domg 6767 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)

Theoremf1oen 6768 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)

Theoremf1dom 6769 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)

Theorembrsdom 6770 Strict dominance relation, meaning " is strictly greater in size than ." Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)

Theoremisfi 6771* Express " is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose " " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)

Theoremenssdom 6772 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)

Theoremdfdom2 6773 Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)

Theoremendom 6774 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)

Theoremsdomdom 6775 Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)

Theoremsdomnen 6776 Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)

Theorembrdom2 6777 Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)

Theorembren2 6778 Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)

Theoremenrefg 6779 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremenref 6780 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)

Theoremeqeng 6781 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)

Theoremdomrefg 6782 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)

Theoremen2d 6783* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)

Theoremen3d 6784* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)

Theoremen2i 6785* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)

Theoremen3i 6786* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)

Theoremdom2lem 6787* 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.)

Theoremdom2d 6788* 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.)

Theoremdom3d 6789* 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.)

Theoremdom2 6790* 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.)

Theoremdom3 6791* 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.)

Theoremidssen 6792 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremssdomg 6793 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremener 6794 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremensymb 6795 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremensym 6796 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremensymi 6797 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)

Theorementr 6798 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)

Theoremdomtr 6799 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theorementri 6800 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)

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