P. 382 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38101-38200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eldisjeq 38101	Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	eldisjeqi 38102	Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)
Theorem	eldisjeqd 38103	Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	disjres 38104*	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 38105	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 38106	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 38107*	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 38108	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 38109	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 38110	Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴))
Theorem	disjimin 38111	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 38112	Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
Theorem	disjimxrnres 38113	Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)))
Theorem	disjALTV0 38114	The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ∅
Theorem	disjALTVid 38115	The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
⊢ Disj I
Theorem	disjALTVidres 38116	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 38117	The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
Theorem	disjALTVxrnidres 38118	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 38119*	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 38120	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 38121	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))
Theorem	dfantisymrel5 38122*	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 38123*	(Contributed by Peter Mazsa, 25-Jun-2024.)
⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
Theorem	antisymrelressn 38124	(Contributed by Peter Mazsa, 29-Jun-2024.)
⊢ AntisymRel (𝑅 ↾ {𝐴})
21.24.19  Partitions: disjoints on domain quotients
Definition	df-parts 38125	Define the class of all partitions, cf. the comment of df-disjs 38064. 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 38130. 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 38133. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Parts = ( DomainQss ↾ Disjs )
Definition	df-part 38126	Define the partition predicate (read: 𝐴 is a partition by 𝑅). Alternative definition is dfpart2 38129. The binary partition and the partition predicate are the same if 𝐴 and 𝑅 are sets, cf. brpartspart 38133. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-membparts 38127	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 38128	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 38130. Member partition is the conventional meaning of partition (see the notes of df-parts 38125 and dfmembpart2 38130), we generalize the concept in df-parts 38125 and df-part 38126. Member partition and comember equivalence are the same by mpet 38199. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
Theorem	dfpart2 38129	Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	dfmembpart2 38130	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 38131	Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))
Theorem	brparts2 38132	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 38133	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 38134	Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
Theorem	parteq2 38135	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ (𝐴 = 𝐵 → (𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵))
Theorem	parteq12 38136	Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐵))
Theorem	parteq1i 38137	Equality theorem for partition, inference version. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴)
Theorem	parteq1d 38138	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 38139	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 38140	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 38141	The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38239, cf. eldisjim 38144. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
Theorem	disjimi 38142	Every disjoint relation generates equivalent cosets by the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)
⊢ Disj 𝑅    ⇒   ⊢ EqvRel ≀ 𝑅
Theorem	detlem 38143	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 38144	If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38239). Special case of disjim 38141. (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 38145	Alternate form of eldisjim 38144. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ ( ElDisj 𝐴 → EqvRel ∼ 𝐴)
Theorem	eqvrel0 38146	The null class is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ∅
Theorem	det0 38147	The cosets by the null class are in equivalence relation if and only if the null class is disjoint (which it is, see disjALTV0 38114). (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ( Disj ∅ ↔ EqvRel ≀ ∅)
Theorem	eqvrelcoss0 38148	The cosets by the null class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ ∅
Theorem	eqvrelid 38149	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 38150	The cosets by a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ EqvRel ≀ ( I ↾ 𝐴)
Theorem	eqvrel1cossinidres 38151	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 38152	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 38153	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 38154	The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ I
Theorem	detidres 38155	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 38156	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 38157	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 38158*	Lemma for disjdmqseq 38165, partim2 38167 and petlem 38172 via disjlem17 38159, (general version of the former prtlem14 38234). (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 38159*	Lemma for disjdmqseq 38165, partim2 38167 and petlem 38172 via disjlem18 38160, (general version of the former prtlem17 38236). (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 38160*	Lemma for disjdmqseq 38165, partim2 38167 and petlem 38172 via disjlem19 38161, (general version of the former prtlem18 38237). (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 38161*	Lemma for disjdmqseq 38165, partim2 38167 and petlem 38172 via disjdmqs 38164, (general version of the former prtlem19 38238). (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Theorem	disjdmqsss 38162	Lemma for disjdmqseq 38165 via disjdmqs 38164. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqscossss 38163	Lemma for disjdmqseq 38165 via disjdmqs 38164. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))
Theorem	disjdmqs 38164	If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 38167 and petlem 38172 via disjdmqseq 38165. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqseq 38165	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 38166 (which is the closest theorem to the former prter2 38241). Lemma for partim2 38167 and petlem 38172. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	eldisjn0el 38166	Special case of disjdmqseq 38165 (perhaps this is the closest theorem to the former prter2 38241). (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	partim2 38167	Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 38168. Lemma for petlem 38172. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	partim 38168	Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 38167. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
Theorem	partimeq 38169	Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 38039. (Contributed by Peter Mazsa, 25-Dec-2024.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))
Theorem	eldisjlem19 38170*	Special case of disjlem19 38161 (together with membpartlem19 38171, this is former prtlem19 38238). (Contributed by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Theorem	membpartlem19 38171*	Together with disjlem19 38161, this is former prtlem19 38238. (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 38172	If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 38193), or converse function (cf. dfdisjALTV 38073), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 38210. (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 38173	If you can prove disjointness (e.g. disjALTV0 38114, disjALTVid 38115, disjALTVidres 38116, disjALTVxrnidres 38118, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 38073), 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 38174	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 38175	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 38176	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 38177	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 38178	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 38179	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 38150. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)
Theorem	petinidres2 38180	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 38181	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 37809, disjALTVinidres 38117 and eqvrel1cossinidres 38151. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)
Theorem	petxrnidres2 38182	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 38183	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 37811, disjALTVxrnidres 38118 and eqvrel1cossxrnidres 38152. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴)
Theorem	eqvreldisj1 38184*	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj2 38185, eqvreldisj3 38186). (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 38185	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 38186). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.)
⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))
Theorem	eqvreldisj3 38186	The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 8785). (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)))
Theorem	eqvreldisj4 38187	Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.)
⊢ ( EqvRel 𝑅 → Disj (𝑆 ∩ (◡ E ↾ (𝐵 / 𝑅))))
Theorem	eqvreldisj5 38188	Range Cartesian product with converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ ( EqvRel 𝑅 → Disj (𝑆 ⋉ (◡ E ↾ (𝐵 / 𝑅))))
Theorem	eqvrelqseqdisj2 38189	Implication of eqvreldisj2 38185, lemma for The Main Theorem of Equivalences mainer 38194. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → ElDisj 𝐴)
Theorem	fences3 38190	Implication of eqvrelqseqdisj2 38189 and n0eldmqseq 38009, see comment of fences 38204. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Theorem	eqvrelqseqdisj3 38191	Implication of eqvreldisj3 38186, lemma for the Member Partition Equivalence Theorem mpet3 38196. (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by Peter Mazsa, 24-Sep-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴))
Theorem	eqvrelqseqdisj4 38192	Lemma for petincnvepres2 38208. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ∩ (◡ E ↾ 𝐴)))
Theorem	eqvrelqseqdisj5 38193	Lemma for the Partition-Equivalence Theorem pet2 38210. (Contributed by Peter Mazsa, 15-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ⋉ (◡ E ↾ 𝐴)))
Theorem	mainer 38194	The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
Theorem	partimcomember 38195	Partition with general 𝑅 (in addition to the member partition cf. mpet 38199 and mpet2 38200) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
Theorem	mpet3 38196	Member Partition-Equivalence Theorem. Together with mpet 38199 mpet2 38200, mostly in its conventional cpet 38198 and cpet2 38197 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38210 with general 𝑅). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	cpet2 38197	The conventional form of the Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have called disjoint or partition what we call element disjoint or member partition, see also cpet 38198. Together with cpet 38198, mpet 38199 mpet2 38200, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38210 with general 𝑅). (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	cpet 38198	The conventional form of Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have been calling disjoint or partition what we call element disjoint or member partition, see also cpet2 38197. Cf. mpet 38199, mpet2 38200 and mpet3 38196 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 38211 and pet2 38210 for Partition-Equivalence Theorem with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ ( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	mpet 38199	Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 38202. Member partition and comember equivalence relation are the same (or: each element of 𝐴 have equivalent comembers if and only if 𝐴 is a member partition). Together with mpet2 38200, mpet3 38196, and with the conventional cpet 38198 and cpet2 38197, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38210 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
Theorem	mpet2 38200	Member Partition-Equivalence Theorem in a shorter form. Together with mpet 38199 mpet3 38196, mostly in its conventional cpet 38198 and cpet2 38197 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38210 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)