P. 388 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfdisjs2 38701	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I }
Theorem	dfdisjs3 38702*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢∀𝑣∀𝑥((𝑢𝑟𝑥 ∧ 𝑣𝑟𝑥) → 𝑢 = 𝑣)}
Theorem	dfdisjs4 38703*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥}
Theorem	dfdisjs5 38704*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}
Theorem	dfdisjALTV 38705	Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 38696 why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅))
Theorem	dfdisjALTV2 38706	Alternate definition of the disjoint relation predicate, cf. dffunALTV2 38680. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfdisjALTV3 38707*	Alternate definition of the disjoint relation predicate, cf. dffunALTV3 38681. (Contributed by Peter Mazsa, 28-Jul-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅))
Theorem	dfdisjALTV4 38708*	Alternate definition of the disjoint relation predicate, cf. dffunALTV4 38682. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	dfdisjALTV5 38709*	Alternate definition of the disjoint relation predicate, cf. dffunALTV5 38683. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
Theorem	dfeldisj2 38710	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I )
Theorem	dfeldisj3 38711*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ∀𝑥 ∈ (𝑢 ∩ 𝑣)𝑢 = 𝑣)
Theorem	dfeldisj4 38712*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
Theorem	dfeldisj5 38713*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
Theorem	eldisjs 38714	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs2 38715	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs3 38716*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs4 38717*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs5 38718*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))
Theorem	eldisjsdisj 38719	The element of the class of disjoint relations and the disjoint relation predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))
Theorem	eleldisjs 38720	Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
Theorem	eleldisjseldisj 38721	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. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴))
Theorem	disjrel 38722	Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.)
⊢ ( Disj 𝑅 → Rel 𝑅)
Theorem	disjss 38723	Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjssi 38724	Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( Disj 𝐵 → Disj 𝐴)
Theorem	disjssd 38725	Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjeq 38726	Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjeqi 38727	Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( Disj 𝐴 ↔ Disj 𝐵)
Theorem	disjeqd 38728	Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjdmqseqeq1 38729	Lemma for the equality theorem for partition parteq1 38766. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)))
Theorem	eldisjss 38730	Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjssi 38731	Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)
Theorem	eldisjssd 38732	Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjeq 38733	Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	eldisjeqi 38734	Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)
Theorem	eldisjeqd 38735	Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	disjres 38736*	Disjoint restriction. (Contributed by Peter Mazsa, 25-Aug-2023.)
⊢ (Rel 𝑅 → ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅)))
Theorem	eldisjn0elb 38737	Two forms of disjoint elements when the empty set is not an element of the class. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
Theorem	disjxrn 38738	Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I )
Theorem	disjxrnres5 38739*	Disjoint range Cartesian product. (Contributed by Peter Mazsa, 25-Aug-2023.)
⊢ ( Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ 𝑆) ∩ [𝑣](𝑅 ⋉ 𝑆)) = ∅))
Theorem	disjorimxrn 38740	Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆))
Theorem	disjimxrn 38741	Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 15-Dec-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ 𝑆))
Theorem	disjimres 38742	Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴))
Theorem	disjimin 38743	Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆))
Theorem	disjiminres 38744	Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
Theorem	disjimxrnres 38745	Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)))
Theorem	disjALTV0 38746	The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ∅
Theorem	disjALTVid 38747	The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
⊢ Disj I
Theorem	disjALTVidres 38748	The class of identity relations restricted is disjoint. (Contributed by Peter Mazsa, 28-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ( I ↾ 𝐴)
Theorem	disjALTVinidres 38749	The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
Theorem	disjALTVxrnidres 38750	The class of range Cartesian product with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 25-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.)
⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
Theorem	disjsuc 38751*	Disjoint range Cartesian product, special case. (Contributed by Peter Mazsa, 25-Aug-2023.)
⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
21.26.18  Antisymmetry
Definition	df-antisymrel 38752	Define the antisymmetric relation predicate. (Read: 𝑅 is an antisymmetric relation.) (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
Theorem	dfantisymrel4 38753	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))
Theorem	dfantisymrel5 38754*	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	antisymrelres 38755*	(Contributed by Peter Mazsa, 25-Jun-2024.)
⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
Theorem	antisymrelressn 38756	(Contributed by Peter Mazsa, 29-Jun-2024.)
⊢ AntisymRel (𝑅 ↾ {𝐴})
21.26.19  Partitions: disjoints on domain quotients
Definition	df-parts 38757	Define the class of all partitions, cf. the comment of df-disjs 38696. Partitions are disjoints on domain quotients (or: domain quotients restricted to disjoints). This is a more general meaning of partition than we we are familiar with: the conventional meaning of partition (e.g. partition 𝐴 of 𝑋, [Halmos] p. 28: "A partition of 𝑋 is a disjoint collection 𝐴 of non-empty subsets of 𝑋 whose union is 𝑋", or Definition 35, [Suppes] p. 83., cf. https://oeis.org/A000110 38696) is what we call membership partition here, cf. dfmembpart2 38762. The binary partitions relation and the partition predicate are the same, that is, (𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴) if 𝐴 and 𝑅 are sets, cf. brpartspart 38765. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Parts = ( DomainQss ↾ Disjs )
Definition	df-part 38758	Define the partition predicate (read: 𝐴 is a partition by 𝑅). Alternative definition is dfpart2 38761. The binary partition and the partition predicate are the same if 𝐴 and 𝑅 are sets, cf. brpartspart 38765. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-membparts 38759	Define the class of member partition relations on their domain quotients. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
Definition	df-membpart 38760	Define the member partition predicate, or the disjoint restricted element relation on its domain quotient predicate. (Read: 𝐴 is a member partition.) A alternative definition is dfmembpart2 38762. Member partition is the conventional meaning of partition (see the notes of df-parts 38757 and dfmembpart2 38762), we generalize the concept in df-parts 38757 and df-part 38758. Member partition and comember equivalence are the same by mpet 38831. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
Theorem	dfpart2 38761	Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	dfmembpart2 38762	Alternate definition of the conventional membership case of partition. Partition 𝐴 of 𝑋, [Halmos] p. 28: "A partition of 𝑋 is a disjoint collection 𝐴 of non-empty subsets of 𝑋 whose union is 𝑋", or Definition 35, [Suppes] p. 83., cf. https://oeis.org/A000110 . (Contributed by Peter Mazsa, 14-Aug-2021.)
⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Theorem	brparts 38763	Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))
Theorem	brparts2 38764	Binary partitions relation. (Contributed by Peter Mazsa, 30-Dec-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ (dom 𝑅 / 𝑅) = 𝐴)))
Theorem	brpartspart 38765	Binary partition and the partition predicate are the same if 𝐴 and 𝑅 are sets. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴))
Theorem	parteq1 38766	Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
Theorem	parteq2 38767	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ (𝐴 = 𝐵 → (𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵))
Theorem	parteq12 38768	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐵))
Theorem	parteq1i 38769	Equality theorem for partition, inference version. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴)
Theorem	parteq1d 38770	Equality theorem for partition, deduction version. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
Theorem	partsuc2 38771	Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴)
Theorem	partsuc 38772	Property of the partition. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ (((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) Part (suc 𝐴 ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴)
21.26.20  Partition-Equivalence Theorems
Theorem	disjim 38773	The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38872, cf. eldisjim 38776. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
Theorem	disjimi 38774	Every disjoint relation generates equivalent cosets by the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)
⊢ Disj 𝑅    ⇒   ⊢ EqvRel ≀ 𝑅
Theorem	detlem 38775	If a relation is disjoint, then it is equivalent to the equivalent cosets of the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)
⊢ Disj 𝑅    ⇒   ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)
Theorem	eldisjim 38776	If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38872). Special case of disjim 38773. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015) (Revised by Peter Mazsa, 8-Feb-2018.) ( Revised by Peter Mazsa, 23-Sep-2021.)
⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴)
Theorem	eldisjim2 38777	Alternate form of eldisjim 38776. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ ( ElDisj 𝐴 → EqvRel ∼ 𝐴)
Theorem	eqvrel0 38778	The null class is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ∅
Theorem	det0 38779	The cosets by the null class are in equivalence relation if and only if the null class is disjoint (which it is, see disjALTV0 38746). (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj ∅ ↔ EqvRel ≀ ∅)
Theorem	eqvrelcoss0 38780	The cosets by the null class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ ∅
Theorem	eqvrelid 38781	The identity relation is an equivalence relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel I
Theorem	eqvrel1cossidres 38782	The cosets by a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ≀ ( I ↾ 𝐴)
Theorem	eqvrel1cossinidres 38783	The cosets by an intersection with a restricted identity relation are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))
Theorem	eqvrel1cossxrnidres 38784	The cosets by a range Cartesian product with a restricted identity relation are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴))
Theorem	detid 38785	The cosets by the identity relation are in equivalence relation if and only if the identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj I ↔ EqvRel ≀ I )
Theorem	eqvrelcossid 38786	The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ I
Theorem	detidres 38787	The cosets by the restricted identity relation are in equivalence relation if and only if the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj ( I ↾ 𝐴) ↔ EqvRel ≀ ( I ↾ 𝐴))
Theorem	detinidres 38788	The cosets by the intersection with the restricted identity relation are in equivalence relation if and only if the intersection with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)))
Theorem	detxrnidres 38789	The cosets by the range Cartesian product with the restricted identity relation are in equivalence relation if and only if the range Cartesian product with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)))
Theorem	disjlem14 38790*	Lemma for disjdmqseq 38797, partim2 38799 and petlem 38804 via disjlem17 38791, (general version of the former prtlem14 38867). (Contributed by Peter Mazsa, 10-Sep-2021.)
⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
Theorem	disjlem17 38791*	Lemma for disjdmqseq 38797, partim2 38799 and petlem 38804 via disjlem18 38792, (general version of the former prtlem17 38869). (Contributed by Peter Mazsa, 10-Sep-2021.)
⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
Theorem	disjlem18 38792*	Lemma for disjdmqseq 38797, partim2 38799 and petlem 38804 via disjlem19 38793, (general version of the former prtlem18 38870). (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝐵))))
Theorem	disjlem19 38793*	Lemma for disjdmqseq 38797, partim2 38799 and petlem 38804 via disjdmqs 38796, (general version of the former prtlem19 38871). (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Theorem	disjdmqsss 38794	Lemma for disjdmqseq 38797 via disjdmqs 38796. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqscossss 38795	Lemma for disjdmqseq 38797 via disjdmqs 38796. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))
Theorem	disjdmqs 38796	If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 38799 and petlem 38804 via disjdmqseq 38797. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqseq 38797	If a relation is disjoint, its domain quotient is equal to a class if and only if the domain quotient of the cosets by it is equal to the class. General version of eldisjn0el 38798 (which is the closest theorem to the former prter2 38874). Lemma for partim2 38799 and petlem 38804. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	eldisjn0el 38798	Special case of disjdmqseq 38797 (perhaps this is the closest theorem to the former prter2 38874). (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	partim2 38799	Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 38800. Lemma for petlem 38804. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	partim 38800	Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 38799. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)