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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 }
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