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Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeceqoveq 7001* Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Theoremth3qlem1 7002* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremth3qlem2 7003* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremth3qcor 7004* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremth3q 7005* Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovec 7006* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See set.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)

Theoremecovcom 7007* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremecovass 7008* Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremecovdi 7009* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)

2.4.29  The mapping operation

Syntaxcmap 7010 Extend the definition of a class to include the mapping operation. (Read for , "the set of all functions that map from to .)

Syntaxcpm 7011 Extend the definition of a class to include the partial mapping operation. (Read for , "the set of all partial functions that map from to .)

Definitiondf-map 7012* Define the mapping operation or set exponentiation. The set of all functions that map from to is written (see mapval 7022). Many authors write followed by as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g. Definition 10.42 of [TakeutiZaring] p. 95). Other authors show as a prefixed superscript, which is read " pre " (e.g. definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(, ) for our . The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.)

Definitiondf-pm 7013* Define the partial mapping operation. A partial function from to is a function from a subset of to . The set of all partial functions from to is written (see pmvalg 7021). A notation for this operation apparently does not appear in the literature. We use to distinguish it from the less general set exponentiation operation (df-map 7012) . See mapsspm 7039 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)

Theoremmapprc 7014* When is a proper class, the class of all functions mapping to is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)

Theorempmex 7015* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Theoremmapex 7016* The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)

Theoremfnmap 7017 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnpm 7018 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremreldmmap 7019 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremmapvalg 7020* The value of set exponentiation. is the set of all functions that map from to . Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorempmvalg 7021* The value of the partial mapping operation. is the set of all partial functions that map from to . (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremmapval 7022* The value of set exponentiation (inference version). is the set of all functions that map from to . Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)

Theoremelmapg 7023 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremelpmg 7024 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2g 7025 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)

Theoremelpm2r 7026 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)

Theoremelpmi 7027 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)

Theorempmfun 7028 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremelmapex 7029 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremelmapi 7030 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremfpmg 7031 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmss12g 7032 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmresg 7033 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theoremelmap 7034 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)

Theoremmapval2 7035* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Theoremelpm 7036 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2 7037 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremfpm 7038 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremmapsspm 7039 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)

Theorempmsspw 7040 Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremmapsspw 7041 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 7042* Special case of fvmpt 5798 for operator theorems. (Contributed by NM, 27-Nov-2007.)

Theoremmap0e 7043 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 7044 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 7045 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 7046 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 7047* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)

Theoremmapss 7048 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 7049* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremfvdiagfn 7050* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremmapsnconst 7051 Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremmapsncnv 7052* 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 7053* Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremmapsnf1o3 7054* Explicit bijection in the reverse of mapsnf1o2 7053. (Contributed by Stefan O'Rear, 24-Mar-2015.)

2.4.30  Infinite Cartesian products

Syntaxcixp 7055 Extend class notation to include infinite Cartesian products.

Definitiondf-ixp 7056* 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 7057* Eliminate the expression in df-ixp 7056, 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 7058* Membership in an infinite Cartesian product. See df-ixp 7056 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)

Theoremfvixp 7059* Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremixpfn 7060* A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremelixp 7061* Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)

Theoremelixpconst 7062* Membership in an infinite Cartesian product of a constant . (Contributed by NM, 12-Apr-2008.)

Theoremixpconstg 7063* Infinite Cartesian product of a constant . (Contributed by Mario Carneiro, 11-Jan-2015.)

Theoremixpconst 7064* Infinite Cartesian product of a constant . (Contributed by NM, 28-Sep-2006.)

Theoremixpeq1 7065* Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Theoremixpeq1d 7066* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremss2ixp 7067 Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)

Theoremixpeq2 7068 Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Theoremixpeq2dva 7069* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremixpeq2dv 7070* Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremcbvixp 7071* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremcbvixpv 7072* Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfixp 7073 Bound-variable hypothesis builder for indexed cross product. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremnfixp1 7074 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 7075* 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 7076* 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 7077* 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 7078* 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 7079* The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremixpiin 7080* The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremixpint 7081* The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremixp0x 7082 An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)

Theoremixpssmap2g 7083* An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7084 avoids ax-rep 4312. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremixpssmapg 7084* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theorem0elixp 7085 Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)

Theoremixpn0 7086 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 8353. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremixp0 7087 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 8353. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)

Theoremixpssmap 7088* An infinite Cartesian product is a subset of set exponentation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)

Theoremresixp 7089* Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Theoremundifixp 7090* Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)

Theoremmptelixpg 7091* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)

Theoremresixpfo 7092* 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 7093* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremixpsnf1o 7094* A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremmapsnf1o 7095* A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremboxriin 7096* 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 7097* The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.)

2.4.31  Equinumerosity

Syntaxcen 7098 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)

Syntaxcdom 7099 Extend class definition to include the dominance relation (curly less-than-or-equal)

Syntaxcsdm 7100 Extend class definition to include the strict dominance relation (curly less-than)

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