P. 395 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqvrel1cossxrnidres 39401	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 39402	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 39403	The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ I
Theorem	detidres 39404	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 39405	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 39406	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 39407*	Lemma for disjdmqseq 39414, partim2 39416 and petlem 39421 via disjlem17 39408, (general version of the former prtlem14 39505). (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 39408*	Lemma for disjdmqseq 39414, partim2 39416 and petlem 39421 via disjlem18 39409, (general version of the former prtlem17 39507). (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 39409*	Lemma for disjdmqseq 39414, partim2 39416 and petlem 39421 via disjlem19 39410, (general version of the former prtlem18 39508). (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 39410*	Lemma for disjdmqseq 39414, partim2 39416 and petlem 39421 via disjdmqs 39413, (general version of the former prtlem19 39509). (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Theorem	disjdmqsss 39411	Lemma for disjdmqseq 39414 via disjdmqs 39413. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqscossss 39412	Lemma for disjdmqseq 39414 via disjdmqs 39413. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))
Theorem	disjdmqs 39413	If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 39416 and petlem 39421 via disjdmqseq 39414. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
Theorem	disjdmqseq 39414	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 39415 (which is the closest theorem to the former prter2 39512). Lemma for partim2 39416 and petlem 39421. (Contributed by Peter Mazsa, 16-Sep-2021.)
⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	eldisjn0el 39415	Special case of disjdmqseq 39414 (perhaps this is the closest theorem to the former prter2 39512). (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	partim2 39416	Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 39417. Lemma for petlem 39421. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))
Theorem	partim 39417	Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 39416. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
Theorem	partimeq 39418	Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 39270. (Contributed by Peter Mazsa, 25-Dec-2024.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))
Theorem	eldisjlem19 39419*	Special case of disjlem19 39410 (together with membpartlem19 39420, this is former prtlem19 39509). (Contributed by Peter Mazsa, 21-Oct-2021.)
⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Theorem	membpartlem19 39420*	Together with disjlem19 39410, this is former prtlem19 39509. (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 39421	If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. eqvrelqseqdisj5 39453), or converse function (cf. dfdisjALTV 39304), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem pet2 39470. (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 39422	If you can prove disjointness (e.g. disjALTV0 39360, disjALTVid 39361, disjALTVidres 39362, disjALTVxrnidres 39364, search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. dfdisjALTV 39304), 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 39423	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 39424	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 39425	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 39426	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 39427	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 39428	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 39399. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)
Theorem	petinidres2 39429	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 39430	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 39045, disjALTVinidres 39363 and eqvrel1cossinidres 39400. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)
Theorem	petxrnidres2 39431	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 39432	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 39047, disjALTVxrnidres 39364 and eqvrel1cossxrnidres 39401. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴)
Theorem	eqvreldisj1 39433*	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj2 39434, eqvreldisj3 39435). (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 39434	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 39435). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.)
⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅))
Theorem	eqvreldisj3 39435	The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 8781). (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 39436	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 39437	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 39438	Implication of eqvreldisj2 39434, lemma for The Main Theorem of Equivalences mainer 39454. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → ElDisj 𝐴)
Theorem	disjimeldisjdmqs 39439	Disj implies element-disjoint quotient carrier. Supplies the carrier-disjointness half of the Disjs pattern: under Disj 𝑅, the coset family is element-disjoint. (Contributed by Peter Mazsa, 5-Feb-2026.)
⊢ ( Disj 𝑅 → ElDisj (dom 𝑅 / 𝑅))
Theorem	eldisjsim1 39440	An element of the class of disjoint relations is disjoint. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ (𝑅 ∈ Disjs → Disj 𝑅)
Theorem	eldisjsim2 39441	An element of the class of disjoint relations is an element of the class of relations. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )
Theorem	disjsssrels 39442	The class of disjoint relations is a subclass of the class of relations. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ Disjs ⊆ Rels
Theorem	eldisjsim3 39443	Disjs implies element-disjoint quotient carrier. Exports the carrier-disjointness property in the ElDisjs packaging. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ (𝑅 ∈ Disjs → (dom 𝑅 / 𝑅) ∈ ElDisjs )
Theorem	eldisjsim4 39444	Disjs implies element-disjoint range of QMap. Same as eldisjsim3 39443 but expressed using the block-map range ran QMap 𝑅 (often the more modular expression). (Contributed by Peter Mazsa, 15-Feb-2026.)
⊢ (𝑅 ∈ Disjs → ran QMap 𝑅 ∈ ElDisjs )
Theorem	eldisjsim5 39445	Disjs is closed under QMap. If a relation is "disjoint-structured" (Disjs), then its canonical block map is also "disjoint-structured". This is the second "structure level" in Disjs: it expresses that the property is stable under passing to the canonical block map, a theme that mirrors Pet-grade stability at a different axis. (Contributed by Peter Mazsa, 15-Feb-2026.)
⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )
Theorem	eldisjs6 39446	Elementhood in the class of disjoints. A relation 𝑅 is in Disjs iff: it is relation-typed, and its quotient-map QMap 𝑅 is itself disjoint, and its quotient-carrier ran QMap 𝑅 = (dom 𝑅 / 𝑅) lies in ElDisjs (element-disjoint carriers). This is the central "stability-by-decomposition" theorem for Disjs: it explains why Disjs is internally well-behaved without adding an external stability clause. It is the exact template that PetParts imitates: for pet 39471, the analogue of "map layer" is the disjointness of the lifted span, the analogue of "carrier layer" is the block-lift fixpoint (BlockLiftFix), and then adds external grade stability (SucMap ShiftStable) which Disjs does not need. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
Theorem	eldisjs7 39447*	Elementhood in the class of disjoints. 𝑅 ∈ Disjs iff: 𝑅 ∈ Rels, and every 𝑥 belongs to at most one block 𝑢 in the quotient-carrier (dom 𝑅 / 𝑅) (element-disjointness at the carrier), and every block 𝑢 in the quotient-carrier has a unique representative 𝑡 ∈ dom 𝑅 such that 𝑢 = [𝑡]𝑅. Provides the "fully expanded" quantifier characterization of the same decomposition as eldisjs6 39446, but without explicitly mentioning QMap. This is the "E*/E!"" view that is closest in spirit to suc11reg 9576-style injectivity and to the "unique generator per block" narrative. It is also the right contrast-point to older one-line criteria like dfdisjs4 39302 (the "u R x" style), because it makes the carrier and representation discipline explicit and type-safe. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))
Theorem	dfdisjs6 39448	Alternate definition of the class of disjoints (via quotient-map stability). Disjs is the class of relations 𝑟 whose quotient-map QMap 𝑟 is again disjoint and whose induced quotient-carrier is element-disjoint. This is the definitional "stability-by-decomposition" packaging of disjointness: it builds Disjs from two internal layers (i) a carrier-layer constraint and (ii) a map-layer closure constraint. This is deliberately different from "u R x" style definitions: it makes the carrier of blocks and the uniqueness-of-representatives discipline first-class and reusable (via QMap) rather than implicit. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ Disjs = {𝑟 ∈ Rels ∣ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )}
Theorem	dfdisjs7 39449*	Alternate definition of the class of disjoints (via carrier disjointness + unique representatives). Ideology-free normal form of dfdisjs6 39448: "blocks cover their elements" (∃*) and "each block has a unique generator" (∃!), expressed entirely at the quotient-carrier level. Same class as dfdisjs6 39448, but presented in fully expanded ∃* / ∃! form over the quotient-carrier (dom 𝑟 / 𝑟). Makes explicit (a) element-disjointness of the quotient-carrier and (b) unique representative existence for each block. These are exactly the two conditions that rule out type-confusions (blocks vs witnesses) and ensure canonical decomposition. This is the form that best supports analogy arguments with df-petparts 39474 and with successor-style uniqueness patterns. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ Disjs = {𝑟 ∈ Rels ∣ (∀𝑥∃*𝑢 ∈ (dom 𝑟 / 𝑟)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑟 / 𝑟)∃!𝑡 ∈ dom 𝑟 𝑢 = [𝑡]𝑟)}
Theorem	fences3 39450	Implication of eqvrelqseqdisj2 39438 and n0eldmqseq 39240, see comment of fences 39464. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Theorem	eqvrelqseqdisj3 39451	Implication of eqvreldisj3 39435, lemma for the Member Partition Equivalence Theorem mpet3 39456. (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by Peter Mazsa, 24-Sep-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴))
Theorem	eqvrelqseqdisj4 39452	Lemma for petincnvepres2 39468. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ∩ (◡ E ↾ 𝐴)))
Theorem	eqvrelqseqdisj5 39453	Lemma for the Partition-Equivalence Theorem pet2 39470. (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 39454	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 39455	Partition with general 𝑅 (in addition to the member partition cf. mpet 39459 and mpet2 39460) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
Theorem	mpet3 39456	Member Partition-Equivalence Theorem. Together with mpet 39459 mpet2 39460, mostly in its conventional cpet 39458 and cpet2 39457 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39470 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 39457	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 39458. Together with cpet 39458, mpet 39459 mpet2 39460, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39470 with general 𝑅). (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	cpet 39458	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 39457. Cf. mpet 39459, mpet2 39460 and mpet3 39456 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 39471 and pet2 39470 for Partition-Equivalence Theorem with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ ( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	mpet 39459	Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 39462. 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 39460, mpet3 39456, and with the conventional cpet 39458 and cpet2 39457, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39470 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
Theorem	mpet2 39460	Member Partition-Equivalence Theorem in a shorter form. Together with mpet 39459 mpet3 39456, mostly in its conventional cpet 39458 and cpet2 39457 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39470 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
Theorem	mpets2 39461	Member Partition-Equivalence Theorem with binary relations, cf. mpet2 39460. (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ ≀ (◡ E ↾ 𝐴) Ers 𝐴))
Theorem	mpets 39462	Member Partition-Equivalence Theorem in its shortest possible form: it shows that member partitions and comember equivalence relations are literally the same. Cf. pet 39471, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ MembParts = CoMembErs
Theorem	mainpart 39463	Partition with general 𝑅 also imply member partition. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
⊢ (𝑅 Part 𝐴 → MembPart 𝐴)
Theorem	fences 39464	The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 39459) generate a partition of the members. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)
Theorem	fences2 39465	The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 39456) generate a partition of the members, it alo means that (𝑅 ErALTV 𝐴 → ElDisj 𝐴) and that (𝑅 ErALTV 𝐴 → ¬ ∅ ∈ 𝐴). (Contributed by Peter Mazsa, 15-Oct-2021.)
⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Theorem	mainer2 39466	The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 15-Oct-2021.)
⊢ (𝑅 ErALTV 𝐴 → ( CoElEqvRel 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Theorem	mainerim 39467	Every equivalence relation implies equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.)
⊢ (𝑅 ErALTV 𝐴 → CoElEqvRel 𝐴)
Theorem	petincnvepres2 39468	A partition-equivalence theorem with intersection and general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ (( Disj (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ∩ (◡ E ↾ 𝐴)) / (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) / ≀ (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴))
Theorem	petincnvepres 39469	The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 39046. Cf. pet 39471. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝑅 ∩ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ErALTV 𝐴)
Theorem	pet2 39470	Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 39471 and pets 39472) is the main result of my investigation into set theory, see the comment of pet 39471. (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))
Theorem	pet 39471	Partition-Equivalence Theorem with general 𝑅 while preserving the restricted converse epsilon relation of mpet2 39460 (as opposed to petincnvepres 39469). A class is a partition by a range Cartesian product with general 𝑅 and the restricted converse element class if and only if the cosets by the range Cartesian product are in an equivalence relation on it. Cf. br1cossxrncnvepres 39048. This theorem (together with pets 39472 and pet2 39470) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 39459, mpet2 39460 and mpet3 39456 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 39460), i.e., it is not the hypothetical General Partition-Equivalence Theorem gpet ⊢ (𝑅 Part 𝐴 ↔ ≀ 𝑅 ErALTV 𝐴), but this one has a general part that mpet2 39460 lacks: 𝑅, which is sufficient for my future application of set theory, for my purpose outside of set theory. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)
Theorem	pets 39472	Partition-Equivalence Theorem with general 𝑅, with binary relations. This theorem (together with pet 39471 and pet2 39470) is the main result of my investigation into set theory, cf. the comment of pet 39471. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))
Theorem	dmqsblocks 39473*	If the pet 39471 span (𝑅 ⋉ (◡ E ↾ 𝐴)) partitions 𝐴, then every block 𝑢 ∈ 𝐴 is of the form [𝑣] for some 𝑣 that not only lies in the domain but also has at least one internal element 𝑐 and at least one 𝑅-target 𝑏 (cf. also the comments of qseq 39239). It makes explicit that pet 39471 gives active representatives for each block, without ever forcing 𝑣 = 𝑢. (Contributed by Peter Mazsa, 23-Nov-2025.)
⊢ ((dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴 → ∀𝑢 ∈ 𝐴 ∃𝑣 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴))∃𝑏∃𝑐(𝑢 = [𝑣](𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ 𝑐 ∈ 𝑣 ∧ 𝑣𝑅𝑏))
21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers
Definition	df-petparts 39474*	Define the class of partition-side general partition-equivalence spans. ⟨𝑟, 𝑛⟩ ∈ PetParts means: (1) 𝑟 is a set-relation (𝑟 ∈ Rels), and (2) 𝑛 is a membership block-carrier (𝑛 ∈ MembParts), and (3) the block-lift span (𝑟 ⋉ (◡ E ↾ 𝑛)) is a generalized partition on its natural quotient-carrier 𝑛 (i.e. (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛). This is the horizontal feasibility base object on the partition side, expressed in the type-safe Parts language. The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) is included at the definition level so later modular refinements can treat typedness as a first-class component (e.g. intersecting a typedness module with disjointness and equilibrium modules) without repeatedly restating it. In particular, it lets decompositions such as dfpetparts2 39478 be written as clean intersections whose first conjunct is exactly the typedness module ( Rels × MembParts ). (Contributed by Peter Mazsa, 19-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
Definition	df-peters 39475*	Define the class of equivalence-side general partition-equivalence spans. ⟨𝑟, 𝑛⟩ ∈ PetErs means: (1) 𝑟 is a set-relation (𝑟 ∈ Rels), and (2) 𝑛 is a carrier recognized on the equivalence side of membership (𝑛 ∈ CoMembErs), and (3) the coset relation of the lifted span, ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)), is an equivalence relation on its natural quotient with carrier 𝑛 (i.e. ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛). This packages the equivalence-view of the same lifted construction that underlies PetParts. It is designed to be parallel to PetParts so later proofs can freely choose the partition side (Parts) or the equivalence side (Ers) without rebuilding the bridge each time; the identification is provided by petseq 39482 (using typesafepets 39481 and mpets 39462). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 39474: to make typedness a reusable module. (Contributed by Peter Mazsa, 19-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
Definition	df-pet2parts 39476	Define the class of grade- and blocklift-stable partition-side general partition-equivalence spans. It consists of those ⟨𝑟, 𝑛⟩ ∈ PetParts such that ⟨𝑟, 𝑛⟩ remains in PetParts after shifting one grade along SucMap (via ShiftStable). Concretely: ⟨𝑟, 𝑛⟩ ∈ PetParts and there exists a predecessor 𝑚 with suc 𝑚 = 𝑛 such that ⟨𝑟, 𝑚⟩ ∈ PetParts (encoded by SucMap ∘ PetParts inside ShiftStable). I.e., it introduces the external (tower/grade) stability axis. This is the "4th level" for pet 39471 (see dfpet2parts2 39479): beyond (i) carrier membership partition, (ii) disjointness, and (iii) semantic equilibrium, we require (iv) stability under a canonical grade shift. PetParts already enforces disjointness and the quotient-carrier equation for the lifted span (hence semantic equilibrium via dfpetparts2 39478). Pet2Parts adds the external grade (tower) stability axis via df-shiftstable 38988 with SucMap. This (iv) is why we need explicit second-level Pet2Parts, while Disjs typically does not: Disjs already packages its own internal two-step consistency (carrier + map) by dfdisjs6 39448 / dfdisjs7 39449, whereas pet 39471 has an additional grade axis that must be imposed separately. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
Definition	df-pet2ers 39477	Define the class of grade- and blocklift-stable equivalence-side general partition-equivalence spans. The equivalence-side analogue of Pet2Parts: stability of PetErs under one-step grade shift along SucMap. Ensures that the equivalence-side formulation supports the same tower/grade infrastructure as the partition-side formulation. SucMap ShiftStable is the grade axis and does not change the equivalence-vs-partition viewpoint (reinforced by pets2eq 39483). (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
Theorem	dfpetparts2 39478*	Alternate definition of PetParts as typedness + disjoint-span + block-lift equilibrium. This theorem is the key modularization step. It decomposes PetParts into the intersection of three orthogonal modules: (T) typedness: ⟨𝑟, 𝑛⟩ ∈ ( Rels × MembParts ), (D) disjoint-span: (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs, (E) semantic equilibrium: ⟨𝑟, 𝑛⟩ ∈ BlockLiftFix, i.e. the carrier 𝑛 is a fixpoint of the induced block-generation operator. Conceptually, (D) provides the disjointness/quotient discipline for the lifted span, while (E) prevents hidden carrier drift (refinement or coarsening of what counts as a block) by enforcing the fixpoint equation. The point of this theorem is that these constraints can be imposed and reused independently by later constructions, while their intersection recovers the intended Parts-based notion. This mirrors the internal packaging of Disjs (see dfdisjs6 39448 / dfdisjs7 39449): for disjoint relations, the "map layer + carrier layer" decomposition is internal via QMap and ElDisjs; for PetParts, the carrier 𝑛 is an external parameter, so the additional carrier stability must be factored explicitly as BlockLiftFix. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )
Theorem	dfpet2parts2 39479*	Grade stability applied to the decomposed PetParts modules. Pet2Parts is obtained by applying the grade-stability operator SucMap ShiftStable (see df-shiftstable 38988) to the modular intersection from dfpetparts2 39478. This makes the two orthogonal stability axes explicit: (E) semantic stability / equilibrium: BlockLiftFix, (G) grade stability: SucMap ShiftStable, assembled on top of typedness and disjoint-span base modules. This is the principled "extra level" that does not arise for Disjs: disjoint relations already bundle their internal map/carrier consistency via QMap and ElDisjs (see dfdisjs6 39448 / dfdisjs7 39449), while the present construction has an additional external grading axis imposed by the canonical successor map SucMap. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))
Theorem	dfpeters2 39480*	Alternate definition of PetErs in fully modular form. This expands the Ers 𝑛 predicate into: (i) a typedness module ( Rels × CoMembErs ), (ii) an equivalence module for the coset relation ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels, (iii) the corresponding quotient-carrier (domain quotient) equation dom ≀ (...) / ≀ (...) = 𝑛. This is the equivalence-side counterpart of the modular decomposition dfpetparts2 39478 on the partition side. (Contributed by Peter Mazsa, 25-Feb-2026.)
⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})
Theorem	typesafepets 39481	Type-safe pets 39472 scheme. On a membership block-carrier 𝐴 ∈ MembParts, the lifted span (𝑅 ⋉ (◡ E ↾ 𝐴)) yields a generalized partition of 𝐴 iff its coset relation yields an equivalence relation on the same carrier 𝐴. This is the type-safe replacement for the earlier broad pets 39472: it explicitly restricts to carriers where 𝐴 is already known to be a block-family (by MembParts). That removes the standard type-safety objection ("are you equating a quotient-carrier of blocks with raw witnesses?") by construction. It is the key bridge used to identify the partition-side and equivalence-side pet classes (petseq 39482), in complete parallel with the membership bridge mpets 39462. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39482 after typedness is enforced by the "Pet*" definitions. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))
Theorem	petseq 39482	Generalized partition-equivalence identification. The partition-side scheme PetParts and the equivalence-side scheme PetErs define the same class of spans (pairs ⟨𝑟, 𝑛⟩). This plays the same organizational role for lifted spans that mpets 39462 plays for carriers: mpets 39462 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 39482 identifies the corresponding span-level predicates built from Parts and Ers. Unlike the earlier broad pets 39472, the bridge used here is the type-safe span theorem typesafepets 39481, which restricts to membership block-carriers. Since typedness (𝑟 ∈ Rels and the appropriate carrier condition) is now built directly into PetParts and PetErs, this theorem can be used downstream without repeatedly re-establishing basic typing premises. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ PetParts = PetErs
Theorem	pets2eq 39483	Grade-stable generalized partition-equivalence identification. After applying the same grade-stability operator (SucMap ShiftStable) to both sides, the grade-stable pet classes still coincide. Confirms that the grade/tower infrastructure is orthogonal to the partition-vs-equivalence viewpoint: stability is preserved under the PetParts = PetErs identification. This is the level at which we can freely work on whichever side is more convenient (Parts for block discipline, Ers for equivalence reasoning), without changing the stable notion of "pet". (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Parts = Pet2Ers
21.27  Mathbox for Rodolfo Medina
21.27.1  Partitions
Theorem	prtlem60 39484	Lemma for prter3 39513. (Contributed by Rodolfo Medina, 9-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
Theorem	bicomdd 39485	Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜒)))
Theorem	jca2r 39486	Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜓 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ∧ 𝜒)))
Theorem	jca3 39487	Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏))))
Theorem	prtlem70 39488	Lemma for prter3 39513: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
⊢ ((((𝜓 ∧ 𝜂) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃 ∧ 𝜏)))) ∧ 𝜂))
Theorem	ibdr 39489	Reverse of ibd 272. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	prtlem100 39490	Lemma for prter3 39513. (Contributed by Rodolfo Medina, 19-Oct-2010.)
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑))
Theorem	prtlem5 39491*	Lemma for prter1 39510, prter2 39512, prter3 39513 and prtex 39511. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))
Theorem	prtlem80 39492	Lemma for prter2 39512. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
Theorem	brabsb2 39493*	A closed form of brabsb 5505. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	eqbrrdv2 39494*	Other version of eqbrrdiv 5770. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	prtlem9 39495*	Lemma for prter3 39513. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
Theorem	prtlem10 39496*	Lemma for prter3 39513. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))
Theorem	prtlem11 39497	Lemma for prter2 39512. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))
Theorem	prtlem12 39498*	Lemma for prtex 39511 and prter3 39513. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )
Theorem	prtlem13 39499*	Lemma for prter1 39510, prter2 39512, prter3 39513 and prtex 39511. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
Theorem	prtlem16 39500*	Lemma for prtex 39511, prter2 39512 and prter3 39513. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ dom ∼ = ∪ 𝐴