P. 380 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37901-38000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eldisjsdisj 37901	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 37902	Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
Theorem	eleldisjseldisj 37903	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 37904	Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.)
⊢ ( Disj 𝑅 → Rel 𝑅)
Theorem	disjss 37905	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 37906	Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( Disj 𝐵 → Disj 𝐴)
Theorem	disjssd 37907	Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjeq 37908	Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjeqi 37909	Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( Disj 𝐴 ↔ Disj 𝐵)
Theorem	disjeqd 37910	Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjdmqseqeq1 37911	Lemma for the equality theorem for partition parteq1 37948. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)))
Theorem	eldisjss 37912	Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjssi 37913	Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)
Theorem	eldisjssd 37914	Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjeq 37915	Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	eldisjeqi 37916	Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)
Theorem	eldisjeqd 37917	Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	disjres 37918*	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 37919	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 37920	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 37921*	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 37922	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 37923	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 37924	Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴))
Theorem	disjimin 37925	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 37926	Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
Theorem	disjimxrnres 37927	Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)))
Theorem	disjALTV0 37928	The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ∅
Theorem	disjALTVid 37929	The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
⊢ Disj I
Theorem	disjALTVidres 37930	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 37931	The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
Theorem	disjALTVxrnidres 37932	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 37933*	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) = ∅))))
21.24.18  Antisymmetry
Definition	df-antisymrel 37934	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 37935	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))
Theorem	dfantisymrel5 37936*	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 37937*	(Contributed by Peter Mazsa, 25-Jun-2024.)
⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
Theorem	antisymrelressn 37938	(Contributed by Peter Mazsa, 29-Jun-2024.)
⊢ AntisymRel (𝑅 ↾ {𝐴})
21.24.19  Partitions: disjoints on domain quotients
Definition	df-parts 37939	Define the class of all partitions, cf. the comment of df-disjs 37878. 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 ) is what we call membership partition here, cf. dfmembpart2 37944. 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 37947. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Parts = ( DomainQss ↾ Disjs )
Definition	df-part 37940	Define the partition predicate (read: 𝐴 is a partition by 𝑅). Alternative definition is dfpart2 37943. The binary partition and the partition predicate are the same if 𝐴 and 𝑅 are sets, cf. brpartspart 37947. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-membparts 37941	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 37942	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 37944. Member partition is the conventional meaning of partition (see the notes of df-parts 37939 and dfmembpart2 37944), we generalize the concept in df-parts 37939 and df-part 37940. Member partition and comember equivalence are the same by mpet 38013. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
Theorem	dfpart2 37943	Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	dfmembpart2 37944	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 37945	Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))
Theorem	brparts2 37946	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 37947	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 37948	Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
Theorem	parteq2 37949	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ (𝐴 = 𝐵 → (𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵))
Theorem	parteq12 37950	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐵))
Theorem	parteq1i 37951	Equality theorem for partition, inference version. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴)
Theorem	parteq1d 37952	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 37953	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 37954	Property of the partition. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ (((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) Part (suc 𝐴 ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴)
21.24.20  Partition-Equivalence Theorems
Theorem	disjim 37955	The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38053, cf. eldisjim 37958. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
Theorem	disjimi 37956	Every disjoint relation generates equivalent cosets by the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)
⊢ Disj 𝑅    ⇒   ⊢ EqvRel ≀ 𝑅
Theorem	detlem 37957	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 37958	If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38053). Special case of disjim 37955. (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 37959	Alternate form of eldisjim 37958. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ ( ElDisj 𝐴 → EqvRel ∼ 𝐴)
Theorem	eqvrel0 37960	The null class is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ∅
Theorem	det0 37961	The cosets by the null class are in equivalence relation if and only if the null class is disjoint (which it is, see disjALTV0 37928). (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj ∅ ↔ EqvRel ≀ ∅)
Theorem	eqvrelcoss0 37962	The cosets by the null class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ ∅
Theorem	eqvrelid 37963	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 37964	The cosets by a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ≀ ( I ↾ 𝐴)
Theorem	eqvrel1cossinidres 37965	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 37966	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 37967	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 37968	The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ I
Theorem	detidres 37969	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 37970	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 37971	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 37972*	Lemma for disjdmqseq 37979, partim2 37981 and petlem 37986 via disjlem17 37973, (general version of the former prtlem14 38048). (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 37973*	Lemma for disjdmqseq 37979, partim2 37981 and petlem 37986 via disjlem18 37974, (general version of the former prtlem17 38050). (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 37974*	Lemma for disjdmqseq 37979, partim2 37981 and petlem 37986 via disjlem19 37975, (general version of the former prtlem18 38051). (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 37975*	Lemma for disjdmqseq 37979, partim2 37981 and petlem 37986 via disjdmqs 37978, (general version of the former prtlem19 38052). (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Theorem	disjdmqsss 37976	Lemma for disjdmqseq 37979 via disjdmqs 37978. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqscossss 37977	Lemma for disjdmqseq 37979 via disjdmqs 37978. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))
Theorem	disjdmqs 37978	If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 37981 and petlem 37986 via disjdmqseq 37979. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqseq 37979	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 37980 (which is the closest theorem to the former prter2 38055). Lemma for partim2 37981 and petlem 37986. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	eldisjn0el 37980	Special case of disjdmqseq 37979 (perhaps this is the closest theorem to the former prter2 38055). (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	partim2 37981	Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 37982. Lemma for petlem 37986. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	partim 37982	Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 37981. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
Theorem	partimeq 37983	Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 37853. (Contributed by Peter Mazsa, 25-Dec-2024.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))
Theorem	eldisjlem19 37984*	Special case of disjlem19 37975 (together with membpartlem19 37985, this is former prtlem19 38052). (Contributed by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Theorem	membpartlem19 37985*	Together with disjlem19 37975, this is former prtlem19 38052. (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 37986	If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 38007), or converse function (cf. dfdisjALTV 37887), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 38024. (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 37987	If you can prove disjointness (e.g. disjALTV0 37928, disjALTVid 37929, disjALTVidres 37930, disjALTVxrnidres 37932, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 37887), 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 37988	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 37989	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 37990	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 37991	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 37992	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 37993	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 37964. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)
Theorem	petinidres2 37994	Class 𝐴 is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
Theorem	petinidres 37995	A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres 37623, disjALTVinidres 37931 and eqvrel1cossinidres 37965. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)
Theorem	petxrnidres2 37996	Class 𝐴 is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))
Theorem	petxrnidres 37997	A class is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. Cf. br1cossxrnidres 37625, disjALTVxrnidres 37932 and eqvrel1cossxrnidres 37966. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴)
Theorem	eqvreldisj1 37998*	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj2 37999, eqvreldisj3 38000). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 3-Dec-2024.)
⊢ ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
Theorem	eqvreldisj2 37999	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 38000). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.)
⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))
Theorem	eqvreldisj3 38000	The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 8792). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 20-Jun-2019.) (Revised by Peter Mazsa, 19-Sep-2021.)
⊢ ( EqvRel 𝑅 → Disj (◡ E ↾ (𝐴 / 𝑅)))