P. 398 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39701-39800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfeqvrel3 39701*	Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ Rel 𝑅))
Theorem	eleqvrels2 39702	Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	eleqvrels3 39703*	Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))
Theorem	eleqvrelsrel 39704	For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate. (Contributed by Peter Mazsa, 24-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅))
Theorem	elcoeleqvrels 39705	Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
Theorem	elcoeleqvrelsrel 39706	For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴))
Theorem	eqvrelrel 39707	An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
⊢ ( EqvRel 𝑅 → Rel 𝑅)
Theorem	eqvrelrefrel 39708	An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → RefRel 𝑅)
Theorem	eqvrelsymrel 39709	An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → SymRel 𝑅)
Theorem	eqvreltrrel 39710	An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → TrRel 𝑅)
Theorem	eqvrelim 39711	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 39712	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 39713	Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)
Theorem	eqvreleqd 39714	Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvrelsym 39715	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 39716	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 39717	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 39718	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 39719	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 39720	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 39721	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 39722	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 39723	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 39724	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 39725	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 39726	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 39727	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 39728*	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 39729	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)
Theorem	eqvrelcoss4 39730*	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 39731	Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 39729, eqvrelcoss3 39728 and eqvrelcoss4 39730 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Theorem	dfcoeleqvrel 39732	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 39729, eqvrelcoss3 39728 and eqvrelcoss4 39730 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
21.28.15  Redundancy
Definition	df-redunds 39733*	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 39736) is equivalent to satisfying the redundancy predicate (df-redund 39734). (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
Definition	df-redund 39734	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 39736) 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 39735	Define the redundancy operator for propositions, cf. df-redund 39734. (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
Theorem	brredunds 39736	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 39737	For sets, binary relation on the class of all redundant sets (brredunds 39736) is equivalent to satisfying the redundancy predicate (df-redund 39734). (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ 𝐴 Redund ⟨𝐵, 𝐶⟩))
Theorem	redundss3 39738	Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ 𝐷 ⊆ 𝐶    ⇒   ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ → 𝐴 Redund ⟨𝐵, 𝐷⟩)
Theorem	redundeq1 39739	Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ 𝐴 = 𝐷    ⇒   ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩)
Theorem	redundpim3 39740	Implication of redundancy of proposition. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜃 → 𝜒)    ⇒   ⊢ ( redund (𝜑, 𝜓, 𝜒) → redund (𝜑, 𝜓, 𝜃))
Theorem	redundpbi1 39741	Equivalence of redundancy of propositions. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ (𝜑 ↔ 𝜃)    ⇒   ⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ redund (𝜃, 𝜓, 𝜒))
Theorem	refrelsredund4 39742	The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 39619) 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 39743	The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 39619) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
Theorem	refrelsredund3 39744*	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 39620) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩
Theorem	refrelredund4 39745	The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 39621) 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 39746	The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 39621) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
Theorem	refrelredund3 39747*	The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 39622) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
21.28.16  Domain quotients
Definition	df-dmqss 39748*	Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8727) of the relation on its domain is equal to the set. See comments of df-ers 39774 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.)
⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
Definition	df-dmqs 39749	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 39757. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
Theorem	dmqseq 39750	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Theorem	dmqseqi 39751	Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)
Theorem	dmqseqd 39752	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 39753	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
Theorem	dmqseqeq1i 39754	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 39755	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 39756	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 39757	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 39758	The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅)
Theorem	qseq 39759*	The quotient set equal to a class. This theorem is used when a class 𝐴 is identified with a quotient (dom 𝑅 / 𝑅). In such a situation, every element 𝑢 ∈ 𝐴 is an 𝑅-coset [𝑣]𝑅 for some 𝑣 ∈ dom 𝑅, but there is no requirement that the "witness" 𝑣 be equal to its own block [𝑣]𝑅. 𝐴 is a set of blocks (equivalence classes), not a set of raw witnesses. In particular, when (dom 𝑅 / 𝑅) = 𝐴 is read together with a partition hypothesis 𝑅 Part 𝐴 (defined as dfpart2 39898), 𝐴 is being treated as the set of blocks [𝑣]𝑅; it does not assert any fixed-point condition 𝑣 = [𝑣]𝑅 such as would arise from the mistaken reading 𝑢 ∈ 𝐴 ↔ 𝑢 = [𝑢]𝑅. Cf. dmqsblocks 39993. (Contributed by Peter Mazsa, 19-Oct-2018.)
⊢ ((𝐵 / 𝑅) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))
Theorem	n0eldmqseq 39760	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 39761	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 39762	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 39763	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 39764	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 39765	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 39766	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 39767	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 39768	Lemma for erimeq2 39789. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))))
Theorem	releldmqs 39769*	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 39770*	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 39771*	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 39772	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 39773	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)
21.28.17  Equivalence relations on domain quotients
Definition	df-ers 39774	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 8721 "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 39694, dfeqvrels2 39698, dfeqvrels3 39699 and df-eqvrel 39695, dfeqvrel2 39700, dfeqvrel3 39701 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8721. While we acknowledge the need of a domain component, the present df-er 8721 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 39992 and pet 39991). From those theorems follows that the natural domain of equivalence relations is not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 36790), but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 39756), see erimeq 39790 vs. prter3 40033. While I'm sure we need both equivalence relation df-eqvrels 39694 and equivalence relation on domain quotient df-ers 39774, 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 39694 and df-ers 39774 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 39775 of the present df-er 8721. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Ers = ( DomainQss ↾ EqvRels )
Definition	df-erALTV 39775	Equivalence relation with natural domain predicate, see also the comment of df-ers 39774. Alternate definition is dferALTV2 39779. 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 39788. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-comembers 39776	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 39777	Define the comember equivalence relation on the class 𝐴 (or, the restricted coelement equivalence relation on its domain quotient 𝐴.) Alternate definitions are dfcomember2 39784 and dfcomember3 39785. 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 39778	Binary equivalence relation with natural domain, see the comment of df-ers 39774. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))
Theorem	dferALTV2 39779	Equivalence relation with natural domain predicate, see the comment of df-ers 39774. (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 39780	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 39781	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 39782	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 39783	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
Theorem	dfcomember2 39784	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	dfcomember3 39785	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 39786	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 39787	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 39788	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 39789	Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 40033 in a more convenient form , see also erimeq 39790). (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 39790	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 40033 and erimeq2 39789). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))
21.28.18  Functions
Definition	df-funss 39791	Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 39792). It is used only by df-funsALTV 39792. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
Definition	df-funsALTV 39792	Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 39794, ... , dffunsALTV5 39798. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ FunsALTV = ( Funss ∩ Rels )
Definition	df-funALTV 39793	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 6541, are always the same, that is ( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 39809. 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 39808. Alternate definitions are dffunALTV2 39799, ... , dffunALTV5 39802. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
Theorem	dffunsALTV 39794	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
Theorem	dffunsALTV2 39795	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }
Theorem	dffunsALTV3 39796*	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 39797*	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 39798*	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 39799	Alternate definition of the function relation predicate, cf. dfdisjALTV2 39825. (Contributed by Peter Mazsa, 8-Feb-2018.)
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
Theorem	dffunALTV3 39800*	Alternate definition of the function relation predicate, cf. dfdisjALTV3 39826. Reproduction of dffun2 6549. 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))