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	dfeldisj4 38701*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
Theorem	dfeldisj5 38702*	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 38703	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs2 38704	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs3 38705*	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 38706*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs5 38707*	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 38708	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 38709	Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
Theorem	eleldisjseldisj 38710	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 38711	Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.)
⊢ ( Disj 𝑅 → Rel 𝑅)
Theorem	disjss 38712	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 38713	Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( Disj 𝐵 → Disj 𝐴)
Theorem	disjssd 38714	Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjeq 38715	Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjeqi 38716	Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( Disj 𝐴 ↔ Disj 𝐵)
Theorem	disjeqd 38717	Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjdmqseqeq1 38718	Lemma for the equality theorem for partition parteq1 38755. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)))
Theorem	eldisjss 38719	Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjssi 38720	Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)
Theorem	eldisjssd 38721	Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjeq 38722	Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	eldisjeqi 38723	Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)
Theorem	eldisjeqd 38724	Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	disjres 38725*	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 38726	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 38727	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 38728*	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 38729	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 38730	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 38731	Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴))
Theorem	disjimin 38732	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 38733	Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
Theorem	disjimxrnres 38734	Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)))
Theorem	disjALTV0 38735	The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ∅
Theorem	disjALTVid 38736	The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
⊢ Disj I
Theorem	disjALTVidres 38737	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 38738	The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
Theorem	disjALTVxrnidres 38739	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 38740*	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 38741	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 38742	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))
Theorem	dfantisymrel5 38743*	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 38744*	(Contributed by Peter Mazsa, 25-Jun-2024.)
⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
Theorem	antisymrelressn 38745	(Contributed by Peter Mazsa, 29-Jun-2024.)
⊢ AntisymRel (𝑅 ↾ {𝐴})
21.26.19  Partitions: disjoints on domain quotients
Definition	df-parts 38746	Define the class of all partitions, cf. the comment of df-disjs 38685. 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 38685) is what we call membership partition here, cf. dfmembpart2 38751. 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 38754. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Parts = ( DomainQss ↾ Disjs )
Definition	df-part 38747	Define the partition predicate (read: 𝐴 is a partition by 𝑅). Alternative definition is dfpart2 38750. The binary partition and the partition predicate are the same if 𝐴 and 𝑅 are sets, cf. brpartspart 38754. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-membparts 38748	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 38749	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 38751. Member partition is the conventional meaning of partition (see the notes of df-parts 38746 and dfmembpart2 38751), we generalize the concept in df-parts 38746 and df-part 38747. Member partition and comember equivalence are the same by mpet 38820. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
Theorem	dfpart2 38750	Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	dfmembpart2 38751	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 38752	Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))
Theorem	brparts2 38753	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 38754	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 38755	Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
Theorem	parteq2 38756	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ (𝐴 = 𝐵 → (𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵))
Theorem	parteq12 38757	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐵))
Theorem	parteq1i 38758	Equality theorem for partition, inference version. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴)
Theorem	parteq1d 38759	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 38760	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 38761	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 38762	The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38860, cf. eldisjim 38765. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
Theorem	disjimi 38763	Every disjoint relation generates equivalent cosets by the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)
⊢ Disj 𝑅    ⇒   ⊢ EqvRel ≀ 𝑅
Theorem	detlem 38764	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 38765	If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38860). Special case of disjim 38762. (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 38766	Alternate form of eldisjim 38765. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ ( ElDisj 𝐴 → EqvRel ∼ 𝐴)
Theorem	eqvrel0 38767	The null class is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ∅
Theorem	det0 38768	The cosets by the null class are in equivalence relation if and only if the null class is disjoint (which it is, see disjALTV0 38735). (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj ∅ ↔ EqvRel ≀ ∅)
Theorem	eqvrelcoss0 38769	The cosets by the null class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ ∅
Theorem	eqvrelid 38770	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 38771	The cosets by a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ≀ ( I ↾ 𝐴)
Theorem	eqvrel1cossinidres 38772	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 38773	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 38774	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 38775	The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ I
Theorem	detidres 38776	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 38777	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 38778	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 38779*	Lemma for disjdmqseq 38786, partim2 38788 and petlem 38793 via disjlem17 38780, (general version of the former prtlem14 38855). (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 38780*	Lemma for disjdmqseq 38786, partim2 38788 and petlem 38793 via disjlem18 38781, (general version of the former prtlem17 38857). (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 38781*	Lemma for disjdmqseq 38786, partim2 38788 and petlem 38793 via disjlem19 38782, (general version of the former prtlem18 38858). (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 38782*	Lemma for disjdmqseq 38786, partim2 38788 and petlem 38793 via disjdmqs 38785, (general version of the former prtlem19 38859). (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Theorem	disjdmqsss 38783	Lemma for disjdmqseq 38786 via disjdmqs 38785. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqscossss 38784	Lemma for disjdmqseq 38786 via disjdmqs 38785. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))
Theorem	disjdmqs 38785	If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 38788 and petlem 38793 via disjdmqseq 38786. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqseq 38786	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 38787 (which is the closest theorem to the former prter2 38862). Lemma for partim2 38788 and petlem 38793. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	eldisjn0el 38787	Special case of disjdmqseq 38786 (perhaps this is the closest theorem to the former prter2 38862). (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	partim2 38788	Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 38789. Lemma for petlem 38793. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	partim 38789	Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 38788. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
Theorem	partimeq 38790	Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 38660. (Contributed by Peter Mazsa, 25-Dec-2024.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))
Theorem	eldisjlem19 38791*	Special case of disjlem19 38782 (together with membpartlem19 38792, this is former prtlem19 38859). (Contributed by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Theorem	membpartlem19 38792*	Together with disjlem19 38782, this is former prtlem19 38859. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → ( MembPart 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Theorem	petlem 38793	If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 38814), or converse function (cf. dfdisjALTV 38694), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 38831. (Contributed by Peter Mazsa, 18-Sep-2021.)
⊢ (( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) → Disj 𝑅)    ⇒   ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	petlemi 38794	If you can prove disjointness (e.g. disjALTV0 38735, disjALTVid 38736, disjALTVidres 38737, disjALTVxrnidres 38739, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 38694), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. (Contributed by Peter Mazsa, 18-Sep-2021.)
⊢ Disj 𝑅    ⇒   ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	pet02 38795	Class 𝐴 is a partition by the null class if and only if the cosets by the null class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴) ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴))
Theorem	pet0 38796	Class 𝐴 is a partition by the null class if and only if the cosets by the null class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (∅ Part 𝐴 ↔ ≀ ∅ ErALTV 𝐴)
Theorem	petid2 38797	Class 𝐴 is a partition by the identity class if and only if the cosets by the identity class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj I ∧ (dom I / I ) = 𝐴) ↔ ( EqvRel ≀ I ∧ (dom ≀ I / ≀ I ) = 𝐴))
Theorem	petid 38798	A class is a partition by the identity class if and only if the cosets by the identity class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( I Part 𝐴 ↔ ≀ I ErALTV 𝐴)
Theorem	petidres2 38799	Class 𝐴 is a partition by the identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
Theorem	petidres 38800	A class is a partition by identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it, cf. eqvrel1cossidres 38771. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)