P. 387 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38601-38700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqvreltrrel 38601	An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → TrRel 𝑅)
Theorem	eqvrelim 38602	Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)
Theorem	eqvreleq 38603	Equality theorem for equivalence relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvreleqi 38604	Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)
Theorem	eqvreleqd 38605	Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvrelsym 38606	An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐴)
Theorem	eqvrelsymb 38607	An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
Theorem	eqvreltr 38608	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 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Theorem	eqvreltrd 38609	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqvreltr4d 38610	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqvrelref 38611	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)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐴)
Theorem	eqvrelth 38612	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)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Theorem	eqvrelcl 38613	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)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
Theorem	eqvrelthi 38614	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)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Theorem	eqvreldisj 38615	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) = ∅))
Theorem	qsdisjALTV 38616	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)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅))    ⇒   ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))
Theorem	eqvrelqsel 38617	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 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)
Theorem	eqvrelcoss 38618	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 20-Dec-2021.)
⊢ ( EqvRel ≀ 𝑅 ↔ TrRel ≀ 𝑅)
Theorem	eqvrelcoss3 38619*	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.)
⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
Theorem	eqvrelcoss2 38620	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)
Theorem	eqvrelcoss4 38621*	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 30-Sep-2021.)
⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
Theorem	dfcoeleqvrels 38622	Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 38620, eqvrelcoss3 38619 and eqvrelcoss4 38621 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Theorem	dfcoeleqvrel 38623	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 38620, eqvrelcoss3 38619 and eqvrelcoss4 38621 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
21.26.13  Redundancy
Definition	df-redunds 38624*	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 38627) is equivalent to satisfying the redundancy predicate (df-redund 38625). (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
Definition	df-redund 38625	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 38627) is equivalent to satisfying the redundancy predicate. (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
Definition	df-redundp 38626	Define the redundancy operator for propositions, cf. df-redund 38625. (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
Theorem	brredunds 38627	Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))
Theorem	brredundsredund 38628	For sets, binary relation on the class of all redundant sets (brredunds 38627) is equivalent to satisfying the redundancy predicate (df-redund 38625). (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ 𝐴 Redund ⟨𝐵, 𝐶⟩))
Theorem	redundss3 38629	Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ 𝐷 ⊆ 𝐶    ⇒   ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ → 𝐴 Redund ⟨𝐵, 𝐷⟩)
Theorem	redundeq1 38630	Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ 𝐴 = 𝐷    ⇒   ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩)
Theorem	redundpim3 38631	Implication of redundancy of proposition. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜃 → 𝜒)    ⇒   ⊢ ( redund (𝜑, 𝜓, 𝜒) → redund (𝜑, 𝜓, 𝜃))
Theorem	redundpbi1 38632	Equivalence of redundancy of propositions. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ (𝜑 ↔ 𝜃)    ⇒   ⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ redund (𝜃, 𝜓, 𝜒))
Theorem	refrelsredund4 38633	The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38514) if the relations are symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
Theorem	refrelsredund2 38634	The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38514) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
Theorem	refrelsredund3 38635*	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 38515) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩
Theorem	refrelredund4 38636	The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38516) 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))
Theorem	refrelredund2 38637	The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38516) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
Theorem	refrelredund3 38638*	The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 38517) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
21.26.14  Domain quotients
Definition	df-dmqss 38639*	Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8751) of the relation on its domain is equal to the set. See comments of df-ers 38664 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.)
⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
Definition	df-dmqs 38640	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 38648. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
Theorem	dmqseq 38641	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Theorem	dmqseqi 38642	Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)
Theorem	dmqseqd 38643	Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Theorem	dmqseqeq1 38644	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
Theorem	dmqseqeq1i 38645	Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)
Theorem	dmqseqeq1d 38646	Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
Theorem	brdmqss 38647	The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	brdmqssqs 38648	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴))
Theorem	n0eldmqs 38649	The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅)
Theorem	n0eldmqseq 38650	The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.)
⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
Theorem	n0elim 38651	Implication of that the empty set is not an element of a class. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
Theorem	n0el3 38652	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)
Theorem	cnvepresdmqss 38653	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.)
⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))
Theorem	cnvepresdmqs 38654	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)
Theorem	unidmqs 38655	The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅))
Theorem	unidmqseq 38656	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.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → (∪ (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))
Theorem	dmqseqim 38657	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.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴)))
Theorem	dmqseqim2 38658	Lemma for erimeq2 38679. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))))
Theorem	releldmqs 38659*	Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)))
Theorem	eldmqs1cossres 38660*	Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
Theorem	releldmqscoss 38661*	Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))
Theorem	dmqscoelseq 38662	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)
Theorem	dmqs1cosscnvepreseq 38663	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)
21.26.15  Equivalence relations on domain quotients
Definition	df-ers 38664	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 8745 "is not standard", "somewhat cryptic", has no constant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. Definitions df-eqvrels 38585, dfeqvrels2 38589, dfeqvrels3 38590 and df-eqvrel 38586, dfeqvrel2 38591, dfeqvrel3 38592 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8745. While we acknowledge the need of a domain component, the present df-er 8745 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 38853 and pet 38852). From those theorems follows that the natural domain of equivalence relations is not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 35936), but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 38647), see erimeq 38680 vs. prter3 38883. While I'm sure we need both equivalence relation df-eqvrels 38585 and equivalence relation on domain quotient df-ers 38664, 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 38585 and df-ers 38664 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 38665 of the present df-er 8745. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Ers = ( DomainQss ↾ EqvRels )
Definition	df-erALTV 38665	Equivalence relation with natural domain predicate, see also the comment of df-ers 38664. Alternate definition is dferALTV2 38669. 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 38678. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-comembers 38666	Define the class of comember equivalence relations on their domain quotients. (Contributed by Peter Mazsa, 28-Nov-2022.) (Revised by Peter Mazsa, 24-Jul-2023.)
⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
Definition	df-comember 38667	Define the comember equivalence relation on the class 𝐴 (or, the restricted coelement equivalence relation on its domain quotient 𝐴.) Alternate definitions are dfcomember2 38674 and dfcomember3 38675. 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.)
⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
Theorem	brers 38668	Binary equivalence relation with natural domain, see the comment of df-ers 38664. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))
Theorem	dferALTV2 38669	Equivalence relation with natural domain predicate, see the comment of df-ers 38664. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.)
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	erALTVeq1 38670	Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))
Theorem	erALTVeq1i 38671	Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)
Theorem	erALTVeq1d 38672	Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))
Theorem	dfcomember 38673	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
Theorem	dfcomember2 38674	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	dfcomember3 38675	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	eqvreldmqs 38676	Two ways to express comember 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))
Theorem	eqvreldmqs2 38677	Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	brerser 38678	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Ers 𝐴 ↔ 𝑅 ErALTV 𝐴))
Theorem	erimeq2 38679	Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 38883 in a more convenient form , see also erimeq 38680). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))
Theorem	erimeq 38680	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 38883 and erimeq2 38679). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))
21.26.16  Functions
Definition	df-funss 38681	Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 38682). It is used only by df-funsALTV 38682. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
Definition	df-funsALTV 38682	Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 38684, ... , dffunsALTV5 38688. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ FunsALTV = ( Funss ∩ Rels )
Definition	df-funALTV 38683	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 6563, are always the same, that is ( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 38699. 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 38698. Alternate definitions are dffunALTV2 38689, ... , dffunALTV5 38692. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
Theorem	dffunsALTV 38684	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
Theorem	dffunsALTV2 38685	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }
Theorem	dffunsALTV3 38686*	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 ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)}
Theorem	dffunsALTV4 38687*	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}
Theorem	dffunsALTV5 38688*	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)}
Theorem	dffunALTV2 38689	Alternate definition of the function relation predicate, cf. dfdisjALTV2 38715. (Contributed by Peter Mazsa, 8-Feb-2018.)
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
Theorem	dffunALTV3 38690*	Alternate definition of the function relation predicate, cf. dfdisjALTV3 38716. Reproduction of dffun2 6571. 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 ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))
Theorem	dffunALTV4 38691*	Alternate definition of the function relation predicate, cf. dfdisjALTV4 38717. This is dffun6 6574. 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))
Theorem	dffunALTV5 38692*	Alternate definition of the function relation predicate, cf. dfdisjALTV5 38718. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))
Theorem	elfunsALTV 38693	Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV2 38694	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV3 38695*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV4 38696*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV5 38697*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTVfunALTV 38698	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.)
⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))
Theorem	funALTVfun 38699	Our definition of the function predicate df-funALTV 38683 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6563, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)
Theorem	funALTVss 38700	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.)
⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))