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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremerthi 6901 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremerdisj 6902 Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecidsn 6903 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)

Theoremqseq1 6904 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremqseq2 6905 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsg 6906* Closed form of elqs 6907. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremelqs 6907* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsi 6908* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremecelqsg 6909 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecelqsi 6910 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecopqsi 6911 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)

Theoremqsexg 6912 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremqsex 6913 A quotient set exists. (Contributed by NM, 14-Aug-1995.)

Theoremuniqs 6914 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Theoremqsss 6915 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremuniqs2 6916 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremsnec 6917 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecqs 6918 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)

Theoremecid 6919 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremqsid 6920 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremectocld 6921* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremectocl 6922* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremelqsn0 6923 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremecelqsdm 6924 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)

Theoremxpider 6925 A square cross product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremiiner 6926* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremriiner 6927* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremerinxp 6928 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecinxp 6929 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Theoremqsinxp 6930 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremqsdisj 6931 Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)

Theoremqsdisj2 6932* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremqsel 6933 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremuniinqs 6934 Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3991. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)

Theoremqliftlem 6935* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftrel 6936* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel 6937* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel1 6938* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfun 6939* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfund 6940* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfuns 6941* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftf 6942* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftval 6943* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremecoptocl 6944* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Theorem2ecoptocl 6945* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)

Theorem3ecoptocl 6946* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)

Theorembrecop 6947* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Theorembrecop2 6948 Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)

Theoremeroveu 6949* Lemma for erov 6951 and eroprf 6952. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremerovlem 6950* Lemma for erov 6951 and eroprf 6952. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremerov 6951* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf 6952* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremerov2 6953* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremeroprf2 6954* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremecopoveq 6955* This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation (specified by the hypothesis) in terms of its operation . (Contributed by NM, 16-Aug-1995.)

Theoremecopovsym 6956* Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremecopovtrn 6957* Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremecopover 6958* Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremeceqoveq 6959* Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Theoremth3qlem1 6960* 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 6961* 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 6962* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremth3q 6963* 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 6964* 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 6965* 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 6966* 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 6967* 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 6968 Extend the definition of a class to include the mapping operation. (Read for , "the set of all functions that map from to .)

Syntaxcpm 6969 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 6970* Define the mapping operation or set exponentiation. The set of all functions that map from to is written (see mapval 6980). 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 6971* 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 6979). 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 6970) . See mapsspm 6997 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)

Theoremmapprc 6972* 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 6973* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Theoremmapex 6974* 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 6975 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnpm 6976 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremreldmmap 6977 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremmapvalg 6978* 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 6979* 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 6980* 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 6981 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremelpmg 6982 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2g 6983 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)

Theoremelpm2r 6984 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)

Theoremelpmi 6985 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)

Theorempmfun 6986 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremelmapex 6987 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremelmapi 6988 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremfpmg 6989 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmss12g 6990 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmresg 6991 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theoremelmap 6992 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)

Theoremmapval2 6993* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Theoremelpm 6994 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2 6995 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremfpm 6996 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremmapsspm 6997 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)

Theorempmsspw 6998 Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremmapsspw 6999 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 7000* Special case of fvmpt 5759 for operator theorems. (Contributed by NM, 27-Nov-2007.)

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