Home Metamath Proof ExplorerTheorem List (p. 361 of 454) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28705) Hilbert Space Explorer (28706-30228) Users' Mathboxes (30229-45330)

Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqvreltr 36001 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)       (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Theoremeqvreltrd 36002 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theoremeqvreltr4d 36003 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴𝑅𝐶)

Theoremeqvrelref 36004 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴 ∈ dom 𝑅)       (𝜑𝐴𝑅𝐴)

Theoremeqvrelth 36005 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴 ∈ dom 𝑅)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Theoremeqvrelcl 36006 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴 ∈ dom 𝑅)

Theoremeqvrelthi 36007 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.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Theoremeqvreldisj 36008 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.) (Revised by Peter Mazsa, 2-Jun-2019.)
( EqvRel 𝑅 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

TheoremqsdisjALTV 36009 Elements of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) (Revised by Peter Mazsa, 3-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐵 ∈ (𝐴 / 𝑅))    &   (𝜑𝐶 ∈ (𝐴 / 𝑅))       (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))

Theoremeqvrelqsel 36010 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.) (Revised by Peter Mazsa, 28-Dec-2019.)
(( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)

Theoremeqvrelcoss 36011 Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 20-Dec-2021.)
( EqvRel ≀ 𝑅 ↔ TrRel ≀ 𝑅)

Theoremeqvrelcoss3 36012* Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.)
( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremeqvrelcoss2 36013 Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)

Theoremeqvrelcoss4 36014* Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 30-Sep-2021.)
( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))

Theoremdfcoeleqvrels 36015 Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 36013, eqvrelcoss3 36012 and eqvrelcoss4 36014 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Theoremdfcoeleqvrel 36016 Alternate definition of the coelement equivalence relation predicate: a coelement equivalence relation is an equivalence relation on coelements. Other alternate definitions should be based on eqvrelcoss2 36013, eqvrelcoss3 36012 and eqvrelcoss4 36014 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

20.22.13  Redundancy

Definitiondf-redunds 36017* Define the class of all redundant sets 𝑥 with respect to 𝑦 in 𝑧. For sets, binary relation on the class of all redundant sets (brredunds 36020) is equivalent to satisfying the redundancy predicate (df-redund 36018). (Contributed by Peter Mazsa, 23-Oct-2022.)
Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}

Definitiondf-redund 36018 Define the redundancy predicate. Read: 𝐴 is redundant with respect to 𝐵 in 𝐶. For sets, binary relation on the class of all redundant sets (brredunds 36020) is equivalent to satisfying the redundancy predicate. (Contributed by Peter Mazsa, 23-Oct-2022.)
(𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶)))

Definitiondf-redundp 36019 Define the redundancy operator for propositions, cf. df-redund 36018. (Contributed by Peter Mazsa, 23-Oct-2022.)
( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ ((𝜑𝜒) ↔ (𝜓𝜒))))

Theorembrredunds 36020 Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))

Theorembrredundsredund 36021 For sets, binary relation on the class of all redundant sets (brredunds 36020) is equivalent to satisfying the redundancy predicate (df-redund 36018). (Contributed by Peter Mazsa, 25-Oct-2022.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ 𝐴 Redund ⟨𝐵, 𝐶⟩))

Theoremredundss3 36022 Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.)
𝐷𝐶       (𝐴 Redund ⟨𝐵, 𝐶⟩ → 𝐴 Redund ⟨𝐵, 𝐷⟩)

Theoremredundeq1 36023 Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.)
𝐴 = 𝐷       (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩)

Theoremredundpim3 36024 Implication of redundancy of proposition. (Contributed by Peter Mazsa, 26-Oct-2022.)
(𝜃𝜒)       ( redund (𝜑, 𝜓, 𝜒) → redund (𝜑, 𝜓, 𝜃))

Theoremredundpbi1 36025 Equivalence of redundancy of propositions. (Contributed by Peter Mazsa, 25-Oct-2022.)
(𝜑𝜃)       ( redund (𝜑, 𝜓, 𝜒) ↔ redund (𝜃, 𝜓, 𝜒))

Theoremrefrelsredund4 36026 The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35912) if the relations are symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)
{𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩

Theoremrefrelsredund2 36027 The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35912) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
{𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩

Theoremrefrelsredund3 36028* The naive version of the class of reflexive relations {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥} is redundant with respect to the class of reflexive relations (see dfrefrels3 35913) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
{𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩

Theoremrefrelredund4 36029 The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35914) if the relation is symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)
redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))

Theoremrefrelredund2 36030 The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35914) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

Theoremrefrelredund3 36031* The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 35915) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)

20.22.14  Domain quotients

Definitiondf-dmqss 36032* Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8282) of the relation on its domain is equal to the set. See comments of df-ers 36056 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.)
DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}

Definitiondf-dmqs 36033 Define the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.) If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same, see brdmqssqs 36041. (Contributed by Peter Mazsa, 9-Aug-2021.)
(𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)

Theoremdmqseq 36034 Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
(𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Theoremdmqseqi 36035 Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
𝑅 = 𝑆       (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)

Theoremdmqseqd 36036 Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Theoremdmqseqeq1 36037 Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
(𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Theoremdmqseqeq1i 36038 Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
𝑅 = 𝑆       ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)

Theoremdmqseqeq1d 36039 Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Theorembrdmqss 36040 The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.)
((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))

Theorembrdmqssqs 36041 If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same. (Contributed by Peter Mazsa, 14-Aug-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴𝑅 DomainQs 𝐴))

Theoremn0eldmqs 36042 The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
¬ ∅ ∈ (dom 𝑅 / 𝑅)

Theoremn0eldmqseq 36043 The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.)
((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)

Theoremn0el3 36044 Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.)
(¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)

Theoremcnvepresdmqss 36045 The domain quotient binary relation of the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
(𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))

Theoremcnvepresdmqs 36046 The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
(( E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)

Theoremunidmqs 36047 The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)
(𝑅𝑉 → (Rel 𝑅 (dom 𝑅 / 𝑅) = ran 𝑅))

Theoremunidmqseq 36048 The union of the domain quotient of a relation is equal to the class 𝐴 if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 28-Dec-2021.)
(𝑅𝑉 → (Rel 𝑅 → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))

Theoremdmqseqim 36049 If the domain quotient of a relation is equal to the class 𝐴, then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴)))

Theoremdmqseqim2 36050 Lemma for erim2 36070. (Contributed by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅𝐵 𝐴))))

Theoremreleldmqs 36051* Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.)
(𝐴𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)))

Theoremeldmqs1cossres 36052* Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.)
(𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))

Theoremreleldmqscoss 36053* Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021.)
(𝐴𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))

Theoremdmqscoelseq 36054 Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)

Theoremdmqs1cosscnvepreseq 36055 Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)

20.22.15  Equivalence relations on domain quotients

Definitiondf-ers 36056 Define the class of equivalence relations on domain quotients (or: domain quotients restricted to equivalence relations).

The present definition of equivalence relation in set.mm df-er 8276 "is not standard", "somewhat cryptic", has no costant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. The definitions df-eqvrels 35978, dfeqvrels2 35982, dfeqvrels3 35983 and df-eqvrel 35979, dfeqvrel2 35984, dfeqvrel3 35985 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8276. While we acknowledge the need of a domain component, the present df-er 8276 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like ~? pets and ~? pet ). From those theorems follows that the natural domain of equivalence relations is

not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 33510),

but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 36040), see erim 36071 vs. prter3 36177.

While I'm sure we need both equivalence relation df-eqvrels 35978 and equivalence relation on domain quotient df-ers 36056, I'm not sure whether we need a third equivalence relation concept with the present dom 𝑅 = 𝐴 component as well: this needs further investigation. As a default I suppose that these two concepts df-eqvrels 35978 and df-ers 36056 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 36057 of the present df-er 8276. (Contributed by Peter Mazsa, 26-Jun-2021.)

Ers = ( DomainQss ↾ EqvRels )

Definitiondf-erALTV 36057 Equivalence relation with natural domain predicate, see also the comment of df-ers 36056. Alternate definition is dferALTV2 36061. Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets, see brerser 36069. (Contributed by Peter Mazsa, 12-Aug-2021.)
(𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅𝑅 DomainQs 𝐴))

Definitiondf-members 36058 Define the class of membership equivalence relations on their domain quotients. (Contributed by Peter Mazsa, 28-Nov-2022.) (Revised by Peter Mazsa, 24-Jul-2023.)
MembErs = {𝑎 ∣ ≀ ( E ↾ 𝑎) Ers 𝑎}

Definitiondf-member 36059 Define the membership equivalence relation on the class 𝐴 (or, the restricted elementhood equivalence relation on its domain quotient 𝐴.) Alternate definitions are dfmember2 36066 and dfmember3 36067.

Later on, in an application of set theory I make a distinction between the default elementhood concept and a special membership concept: membership equivalence relation will be an integral part of that membership concept. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 28-Nov-2022.)

( MembEr 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)

Theorembrers 36060 Binary equivalence relation with natural domain, see the comment of df-ers 36056. (Contributed by Peter Mazsa, 23-Jul-2021.)
(𝐴𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))

TheoremdferALTV2 36061 Equivalence relation with natural domain predicate, see the comment of df-ers 36056. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.)
(𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

TheoremerALTVeq1 36062 Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.)
(𝑅 = 𝑆 → (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴))

TheoremerALTVeq1i 36063 Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.)
𝑅 = 𝑆       (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴)

TheoremerALTVeq1d 36064 Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴))

Theoremdfmember 36065 Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Theoremdfmember2 36066 Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
( MembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))

Theoremdfmember3 36067 Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
( MembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Theoremeqvreldmqs 36068 Two ways to express membership equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
(( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Theorembrerser 36069 Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets. (Contributed by Peter Mazsa, 25-Aug-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 Ers 𝐴𝑅 ErALTV 𝐴))

Theoremerim2 36070 Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 36177 in a more convenient form , see also erim 36071). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))

Theoremerim 36071 Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is the most convenient form of prter3 36177 and erim2 36070). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))

20.22.16  Functions

Definitiondf-funss 36072 Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 36073). It is used only by df-funsALTV 36073. (Contributed by Peter Mazsa, 17-Jul-2021.)
Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }

Definitiondf-funsALTV 36073 Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 36075, ... , dffunsALTV5 36079. (Contributed by Peter Mazsa, 17-Jul-2021.)
FunsALTV = ( Funss ∩ Rels )

Definitiondf-funALTV 36074 Define the function relation predicate, i.e., the function predicate. This definition of the function predicate (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6330, are always the same, that is ( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 36090.

The element of the class of functions and the function predicate are the same, that is (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹) when 𝐹 is a set, see elfunsALTVfunALTV 36089. Alternate definitions are dffunALTV2 36080, ... , dffunALTV5 36083. (Contributed by Peter Mazsa, 17-Jul-2021.)

( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))

TheoremdffunsALTV 36075 Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }

TheoremdffunsALTV2 36076 Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }

TheoremdffunsALTV3 36077* Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀ x1 y1 y2 (( x1 𝑓 y1 x1 𝑓 y2 ) → y1 = y2 )}. (Contributed by Peter Mazsa, 30-Aug-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)}

TheoremdffunsALTV4 36078* Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀𝑥1∃*𝑦1𝑥1𝑓𝑦1}. (Contributed by Peter Mazsa, 31-Aug-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}

TheoremdffunsALTV5 36079* Alternate definition of the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}

TheoremdffunALTV2 36080 Alternate definition of the function relation predicate, cf. dfdisjALTV2 36106. (Contributed by Peter Mazsa, 8-Feb-2018.)
( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))

TheoremdffunALTV3 36081* Alternate definition of the function relation predicate, cf. dfdisjALTV3 36107. Reproduction of dffun2 6338. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀ x1 y1 y2 (( x1 𝑓 y1 x1 𝑓 y2 ) → y1 = y2 ) ∧ Rel 𝐹). (Contributed by NM, 29-Dec-1996.)
( FunALTV 𝐹 ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))

TheoremdffunALTV4 36082* Alternate definition of the function relation predicate, cf. dfdisjALTV4 36108. This is dffun6 6343. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀𝑥1∃*𝑦1𝑥1𝐹𝑦1 ∧ Rel 𝐹). (Contributed by NM, 9-Mar-1995.)
( FunALTV 𝐹 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))

TheoremdffunALTV5 36083* Alternate definition of the function relation predicate, cf. dfdisjALTV5 36109. (Contributed by Peter Mazsa, 5-Sep-2021.)
( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ Rel 𝐹))

TheoremelfunsALTV 36084 Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
(𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTV2 36085 Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTV3 36086* Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐹 ∈ FunsALTV ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTV4 36087* Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥𝐹 ∈ Rels ))

TheoremelfunsALTV5 36088* Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021.)
(𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTVfunALTV 36089 The element of the class of functions and the function predicate are the same when 𝐹 is a set. (Contributed by Peter Mazsa, 26-Jul-2021.)
(𝐹𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))

TheoremfunALTVfun 36090 Our definition of the function predicate df-funALTV 36074 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6330, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
( FunALTV 𝐹 ↔ Fun 𝐹)

TheoremfunALTVss 36091 Subclass theorem for function. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) (Revised by Peter Mazsa, 22-Sep-2021.)
(𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))

TheoremfunALTVeq 36092 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
(𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

TheoremfunALTVeqi 36093 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 = 𝐵       ( FunALTV 𝐴 ↔ FunALTV 𝐵)

TheoremfunALTVeqd 36094 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

20.22.17  Disjoints vs. converse functions

Definitiondf-disjss 36095 Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 36096). It is used only by df-disjs 36096. (Contributed by Peter Mazsa, 17-Jul-2021.)
Disjss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }

Definitiondf-disjs 36096 Define the disjoint relations class, i.e., the class of disjoints. We need Disjs for the definition of Parts and Part for the Partition-Equivalence Theorems: this need for Parts as disjoint relations on their domain quotients is the reason why we must define Disjs instead of simply using converse functions (cf. dfdisjALTV 36105).

The element of the class of disjoints and the disjoint predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set, see eldisjsdisj 36119. Alternate definitions are dfdisjs 36100, ... , dfdisjs5 36104. (Contributed by Peter Mazsa, 17-Jul-2021.)

Disjs = ( Disjss ∩ Rels )

Definitiondf-disjALTV 36097 Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 36105, see the comment of df-disjs 36096 why we need disjoint relations instead of converse functions anyway.

The element of the class of disjoints and the disjoint predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set, see eldisjsdisj 36119. Alternate definitions are dfdisjALTV 36105, ... , dfdisjALTV5 36109. (Contributed by Peter Mazsa, 17-Jul-2021.)

( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))

Definitiondf-eldisjs 36098 Define the disjoint elementhood relations class, i.e., the disjoint elements class. The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set, see eleldisjseldisj 36121. (Contributed by Peter Mazsa, 28-Nov-2022.)
ElDisjs = {𝑎 ∣ ( E ↾ 𝑎) ∈ Disjs }

Definitiondf-eldisj 36099 Define the disjoint elementhood relation predicate, i.e., the disjoint elementhood predicate. Read: the elements of 𝐴 are disjoint. The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set, see eleldisjseldisj 36121.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 36167 with dfeldisj5 36113. See also the comments of ~? dfmembpart2 and of ~? df-parts . (Contributed by Peter Mazsa, 17-Jul-2021.)

( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))

Theoremdfdisjs 36100 Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.)
Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }

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 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45330
 Copyright terms: Public domain < Previous  Next >