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

Theoremerexb 6701 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremiserd 6702* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theorembrdifun 6703 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremswoer 6704* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremswoord1 6705* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoord2 6706* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoso 6707* If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremeqerlem 6708* Lemma for eqer 6709. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremeqer 6709* Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremider 6710 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theorem0er 6711 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremeceq1 6712 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremeceq2 6713 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremelecg 6714 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremelec 6715 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)

Theoremrelelec 6716 Membership in an equivalence class when is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremecss 6717 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecdmn0 6718 A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremereldm 6719 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerth 6720 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremerth2 6721 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremerthi 6722 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 6723 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 6724 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)

Theoremqseq1 6725 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremqseq2 6726 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsg 6727* Closed form of elqs 6728. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremelqs 6728* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsi 6729* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremecelqsg 6730 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecelqsi 6731 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecopqsi 6732 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)

Theoremqsexg 6733 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremqsex 6734 A quotient set exists. (Contributed by NM, 14-Aug-1995.)

Theoremuniqs 6735 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Theoremqsss 6736 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 6737 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremsnec 6738 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecqs 6739 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)

Theoremecid 6740 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 6741 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 6742* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremectocl 6743* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremelqsn0 6744 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremecelqsdm 6745 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)

Theoremxpider 6746 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 6747* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremriiner 6748* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremerinxp 6749 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecinxp 6750 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Theoremqsinxp 6751 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremqsdisj 6752 Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)

Theoremqsdisj2 6753* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremqsel 6754 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.)

Theoremqliftlem 6755* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftrel 6756* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel 6757* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel1 6758* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfun 6759* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfund 6760* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfuns 6761* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftf 6762* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftval 6763* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremecoptocl 6764* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Theorem2ecoptocl 6765* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)

Theorem3ecoptocl 6766* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)

Theorembrecop 6767* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Theorembrecop2 6768 Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)

Theoremeroveu 6769* Lemma for erov 6771 and eroprf 6772. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremerovlem 6770* Lemma for erov 6771 and eroprf 6772. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremerov 6771* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf 6772* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremerov2 6773* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremeroprf2 6774* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremecopoveq 6775* 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 6776* 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 6777* 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 6778* 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 6779* Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Theoremth3qlem1 6780* 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 6781* 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 6782* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremth3q 6783* 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 6784* 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 6785* 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 6786* 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 6787* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)

2.4.28  The mapping operation

Syntaxcmap 6788 Extend the definition of a class to include the mapping operation. (Read for , "the set of all functions that map from to .)

Syntaxcpm 6789 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 6790* Define the mapping operation or set exponentiation. The set of all functions that map from to is written (see mapval 6800). 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 6791* 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 6799). 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 6790) . See mapsspm 6817 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)

Theoremmapprc 6792* 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 6793* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Theoremmapex 6794* 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 6795 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnpm 6796 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremreldmmap 6797 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremmapvalg 6798* 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 6799* 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 6800* 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.)

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