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	refsymrels2 38601	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 38624) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38549, cf. the comment of dfrefrels2 38549. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
Theorem	refsymrels3 38602*	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 38625) can use the ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) version of dfrefrels3 38550, cf. the comment of dfrefrel3 38552. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))}
Theorem	refsymrel2 38603	A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrel2 38551, cf. the comment of dfrefrels2 38549. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	refsymrel3 38604*	A relation which is reflexive and symmetric (like an equivalence relation) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for its reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrel3 38552, cf. the comment of dfrefrel3 38552. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅))
Theorem	elrefsymrels2 38605	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 38624) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38549, cf. the comment of dfrefrels2 38549. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrels3 38606*	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 38625) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrels3 38550, cf. the comment of dfrefrel3 38552. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrelsrel 38607	For sets, being an element of the class of reflexive and symmetric relations is equivalent to satisfying the reflexive and symmetric relation predicates. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
21.26.11  Transitivity
Definition	df-trs 38608	Define the class of all transitive sets (versus the transitive class defined in df-tr 5199). It is used only by df-trrels 38609. Note the similarity of the definitions of df-refs 38546, df-syms 38578 and df-trs 38608. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-trrels 38609	Define the class of transitive relations. For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38617. Alternate definitions are dftrrels2 38611 and dftrrels3 38612. This definition is similar to the definitions of the classes of reflexive (df-refrels 38547) and symmetric (df-symrels 38579) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ TrRels = ( Trs ∩ Rels )
Definition	df-trrel 38610	Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 38609) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38617. Alternate definitions are dftrrel2 38613 and dftrrel3 38614. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dftrrels2 38611	Alternate definition of the class of transitive relations. I'd prefer to define the class of transitive relations by using the definition of composition by [Suppes] p. 63. df-coSUP (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐴𝑢 ∧ 𝑢𝐵𝑦)} as opposed to the present definition of composition df-co 5625 (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐵𝑢 ∧ 𝑢𝐴𝑦)} because the Suppes definition keeps the order of 𝐴, 𝐵, 𝐶, 𝑅, 𝑆, 𝑇 by default in trsinxpSUP (((𝑅 ∩ (𝐴 × 𝐵)) ∘ (𝑆 ∩ (𝐵 × 𝐶))) ⊆ (𝑇 ∩ (𝐴 × 𝐶)) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀ 𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧)) while the present definition of composition disarranges them: trsinxp (((𝑆 ∩ (𝐵 × 𝐶)) ∘ (𝑅 ∩ (𝐴 × 𝐵))) ⊆ (𝑇 ∩ (𝐴 × 𝐶 )) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧) ). This is not mission critical to me, the implication of the Suppes definition is just more aesthetic, at least in the above case. If we swap to the Suppes definition of class composition, I would define the present class of all transitive sets as df-trsSUP and I would consider to switch the definition of the class of cosets by 𝑅 from the present df-coss 38447 to a df-cossSUP. But perhaps there is a mathematical reason to keep the present definition of composition. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
Theorem	dftrrels3 38612*	Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
Theorem	dftrrel2 38613	Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
Theorem	dftrrel3 38614*	Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ ( TrRel 𝑅 ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
Theorem	eltrrels2 38615	Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	eltrrels3 38616*	Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ TrRels ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))
Theorem	eltrrelsrel 38617	For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))
Theorem	trreleq 38618	Equality theorem for the transitive relation predicate. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
Theorem	trrelressn 38619	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38478) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.)
⊢ TrRel (𝑅 ↾ {𝐴})
21.26.12  Equivalence relations
Definition	df-eqvrels 38620	Define the class of equivalence relations. For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 38630. Alternate definitions are dfeqvrels2 38624 and dfeqvrels3 38625. (Contributed by Peter Mazsa, 7-Nov-2018.)
⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
Definition	df-eqvrel 38621	Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 38620) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 38630. Alternate definitions are dfeqvrel2 38626 and dfeqvrel3 38627. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
Definition	df-coeleqvrels 38622	Define the coelement equivalence relations class, the class of sets with coelement equivalence relations. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38632. Alternate definition is dfcoeleqvrels 38657. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
Definition	df-coeleqvrel 38623	Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38658. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38632. (Contributed by Peter Mazsa, 11-Dec-2021.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
Theorem	dfeqvrels2 38624	Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
Theorem	dfeqvrels3 38625*	Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
Theorem	dfeqvrel2 38626	Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	dfeqvrel3 38627*	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 38628	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 38629*	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 38630	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 38631	Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
Theorem	elcoeleqvrelsrel 38632	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 38633	An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
⊢ ( EqvRel 𝑅 → Rel 𝑅)
Theorem	eqvrelrefrel 38634	An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → RefRel 𝑅)
Theorem	eqvrelsymrel 38635	An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → SymRel 𝑅)
Theorem	eqvreltrrel 38636	An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → TrRel 𝑅)
Theorem	eqvrelim 38637	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 38638	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 38639	Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)
Theorem	eqvreleqd 38640	Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvrelsym 38641	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 38642	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 38643	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 38644	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 38645	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 38646	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 38647	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 38648	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 38649	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 38650	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 38651	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 38652	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 38653	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 38654*	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 38655	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)
Theorem	eqvrelcoss4 38656*	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 38657	Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 38655, eqvrelcoss3 38654 and eqvrelcoss4 38656 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Theorem	dfcoeleqvrel 38658	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 38655, eqvrelcoss3 38654 and eqvrelcoss4 38656 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
21.26.13  Redundancy
Definition	df-redunds 38659*	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 38662) is equivalent to satisfying the redundancy predicate (df-redund 38660). (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
Definition	df-redund 38660	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 38662) 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 38661	Define the redundancy operator for propositions, cf. df-redund 38660. (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
Theorem	brredunds 38662	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 38663	For sets, binary relation on the class of all redundant sets (brredunds 38662) is equivalent to satisfying the redundancy predicate (df-redund 38660). (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ 𝐴 Redund ⟨𝐵, 𝐶⟩))
Theorem	redundss3 38664	Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ 𝐷 ⊆ 𝐶    ⇒   ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ → 𝐴 Redund ⟨𝐵, 𝐷⟩)
Theorem	redundeq1 38665	Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ 𝐴 = 𝐷    ⇒   ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩)
Theorem	redundpim3 38666	Implication of redundancy of proposition. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜃 → 𝜒)    ⇒   ⊢ ( redund (𝜑, 𝜓, 𝜒) → redund (𝜑, 𝜓, 𝜃))
Theorem	redundpbi1 38667	Equivalence of redundancy of propositions. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ (𝜑 ↔ 𝜃)    ⇒   ⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ redund (𝜃, 𝜓, 𝜒))
Theorem	refrelsredund4 38668	The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38549) 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 38669	The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38549) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
Theorem	refrelsredund3 38670*	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 38550) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩
Theorem	refrelredund4 38671	The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38551) 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 38672	The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38551) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
Theorem	refrelredund3 38673*	The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 38552) 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 38674*	Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8628) of the relation on its domain is equal to the set. See comments of df-ers 38700 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.)
⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
Definition	df-dmqs 38675	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 38683. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
Theorem	dmqseq 38676	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Theorem	dmqseqi 38677	Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)
Theorem	dmqseqd 38678	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 38679	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
Theorem	dmqseqeq1i 38680	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 38681	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 38682	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 38683	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 38684	The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅)
Theorem	qseq 38685*	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 38806), 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 38890. (Contributed by Peter Mazsa, 19-Oct-2018.)
⊢ ((𝐵 / 𝑅) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))
Theorem	n0eldmqseq 38686	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 38687	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 38688	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 38689	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 38690	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 38691	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 38692	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 38693	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 38694	Lemma for erimeq2 38715. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))))
Theorem	releldmqs 38695*	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 38696*	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 38697*	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 38698	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 38699	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 38700	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 8622 "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 38620, dfeqvrels2 38624, dfeqvrels3 38625 and df-eqvrel 38621, dfeqvrel2 38626, dfeqvrel3 38627 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8622. While we acknowledge the need of a domain component, the present df-er 8622 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 38889 and pet 38888). From those theorems follows that the natural domain of equivalence relations is not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 35968), but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 38682), see erimeq 38716 vs. prter3 38920. While I'm sure we need both equivalence relation df-eqvrels 38620 and equivalence relation on domain quotient df-ers 38700, 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 38620 and df-ers 38700 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 38701 of the present df-er 8622. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Ers = ( DomainQss ↾ EqvRels )