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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremecelqsi 6601 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecopqsi 6602 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)

Theoremqsexg 6603 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremqsex 6604 A quotient set exists. (Contributed by NM, 14-Aug-1995.)

Theoremuniqs 6605 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Theoremqsss 6606 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 6607 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremsnec 6608 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecqs 6609 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)

Theoremecid 6610 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 6611 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 6612* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremectocl 6613* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremelqsn0 6614 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremecelqsdm 6615 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)

Theoremxpider 6616 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 6617* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremriiner 6618* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremerinxp 6619 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecinxp 6620 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Theoremqsinxp 6621 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremqsdisj 6622 Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)

Theoremqsdisj2 6623* A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremqsel 6624 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 6625* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftrel 6626* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel 6627* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel1 6628* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfun 6629* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfund 6630* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfuns 6631* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftf 6632* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftval 6633* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremecoptocl 6634* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Theorem2ecoptocl 6635* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)

Theorem3ecoptocl 6636* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)

Theorembrecop 6637* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Theorembrecop2 6638 Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)

Theoremeroveu 6639* Lemma for erov 6641 and eroprf 6642. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremerovlem 6640* Lemma for erov 6641 and eroprf 6642. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremerov 6641* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf 6642* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremerov2 6643* The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremeroprf2 6644* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremecopoveq 6645* 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 6646* 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 6647* 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 6648* 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 6649* Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Theoremth3qlem1 6650* 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 6651* 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 6652* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremth3q 6653* 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 6654* 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 6655* 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 6656* 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 6657* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)

2.4.26  The mapping operation

Syntaxcmap 6658 Extend the definition of a class to include the mapping operation. (Read for , "the set of all functions that map from to .)

Syntaxcpm 6659 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 6660* Define the mapping operation or set exponentiation. The set of all functions that map from to is written (see mapval 6670). 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 6661* 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 6669). 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 6660) . See mapsspm 6687 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)

Theoremmapprc 6662* 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 6663* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Theoremmapex 6664* 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 6665 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnpm 6666 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremreldmmap 6667 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremmapvalg 6668* 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 6669* 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 6670* 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 6671 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremelpmg 6672 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2g 6673 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)

Theoremelpm2r 6674 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)

Theoremelpmi 6675 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)

Theorempmfun 6676 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremelmapex 6677 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremelmapi 6678 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremfpmg 6679 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmss12g 6680 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theorempmresg 6681 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Theoremelmap 6682 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)

Theoremmapval2 6683* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Theoremelpm 6684 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2 6685 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremfpm 6686 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)

Theoremmapsspm 6687 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)

Theorempmsspw 6688 Partial maps are a subset of the power set of the cross product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremmapsspw 6689 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 6690* Special case of fvmpt 5454 for operator theorems. (Contributed by NM, 27-Nov-2007.)

Theoremmap0e 6691 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 6692 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 6693 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 6694 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 6695* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)

Theoremmapss 6696 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 6697* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremfvdiagfn 6698* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremmapsnconst 6699 Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremmapsncnv 6700* 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.)

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