Home Intuitionistic Logic ExplorerTheorem List (p. 66 of 136) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqer 6501* Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremider 6502 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 6503 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremeceq1 6504 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremeceq1d 6505 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)

Theoremeceq2 6506 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremelecg 6507 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremelec 6508 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)

Theoremrelelec 6509 Membership in an equivalence class when is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremecss 6510 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecdmn0m 6511* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)

Theoremereldm 6512 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerth 6513 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 6514 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 6515 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.)

Theoremecidsn 6516 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)

Theoremqseq1 6517 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremqseq2 6518 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsg 6519* Closed form of elqs 6520. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremelqs 6520* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsi 6521* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremecelqsg 6522 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecelqsi 6523 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecopqsi 6524 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)

Theoremqsexg 6525 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremqsex 6526 A quotient set exists. (Contributed by NM, 14-Aug-1995.)

Theoremuniqs 6527 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Theoremqsss 6528 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 6529 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremsnec 6530 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecqs 6531 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)

Theoremecid 6532 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.)

Theoremecidg 6533 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)

Theoremqsid 6534 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 6535* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremectocl 6536* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremelqsn0m 6537* An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)

Theoremelqsn0 6538 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremecelqsdm 6539 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)

Theoremxpider 6540 A square Cartesian 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.)

Theoremiinerm 6541* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremriinerm 6542* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremerinxp 6543 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecinxp 6544 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Theoremqsinxp 6545 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremqsel 6546 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 6547* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftrel 6548* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel 6549* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel1 6550* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfun 6551* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfund 6552* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfuns 6553* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftf 6554* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftval 6555* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremecoptocl 6556* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Theorem2ecoptocl 6557* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)

Theorem3ecoptocl 6558* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)

Theorembrecop 6559* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Theoremeroveu 6560* Lemma for eroprf 6562. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremerovlem 6561* Lemma for eroprf 6562. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf 6562* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf2 6563* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremecopoveq 6564* 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 6565* 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 6566* 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 6567* 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.)

Theoremecopovsymg 6568* Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)

Theoremecopovtrng 6569* 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 Jim Kingdon, 1-Sep-2019.)

Theoremecopoverg 6570* 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 Jim Kingdon, 1-Sep-2019.)

Theoremth3qlem1 6571* 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 6572* 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 6573* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremth3q 6574* 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.)

Theoremoviec 6575* Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.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 6576* Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6577 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremecovicom 6577* Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)

Theoremecovass 6578* Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6579 will be more useful. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremecoviass 6579* Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)

Theoremecovdi 6580* Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6581 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Theoremecovidi 6581* Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)

2.6.25  The mapping operation

Syntaxcmap 6582 Extend the definition of a class to include the mapping operation. (Read for , "the set of all functions that map from to .)

Syntaxcpm 6583 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 6584* Define the mapping operation or set exponentiation. The set of all functions that map from to is written (see mapval 6594). 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 6585* 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 6593). 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 6584) . See mapsspm 6616 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)

Theoremmapprc 6586* 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 6587* The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Theoremmapex 6588* 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 6589 Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnpm 6590 Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremreldmmap 6591 Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremmapvalg 6592* 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 6593* 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 6594* 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 6595 Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremelmapd 6596 Deduction form of elmapg 6595. (Contributed by BJ, 11-Apr-2020.)

Theoremmapdm0 6597 The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)

Theoremelpmg 6598 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremelpm2g 6599 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)

Theoremelpm2r 6600 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13593
 Copyright terms: Public domain < Previous  Next >