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Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrneq2 35501 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremxrneq2i 35502 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremxrneq2d 35503 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremxrneq12 35504 Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremxrneq12i 35505 Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremxrneq12d 35506 Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremelecxrn 35507* Elementhood in the (𝑅𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
(𝐴𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))

Theoremecxrn 35508* The (𝑅𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
(𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})

Theoremxrninxp 35509* Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.)
((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩))}

Theoremxrninxp2 35510* Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)
((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}

Theoremxrninxpex 35511 Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)

Theoreminxpxrn 35512 Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.)
((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))

Theorembr1cnvxrn2 35513* The converse of a binary relation over a range Cartesian product. (Contributed by Peter Mazsa, 11-Jul-2021.)
(𝐵𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))

Theoremelec1cnvxrn2 35514* Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.)
(𝐵𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))

Theoremrnxrn 35515* Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
ran (𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑆𝑦)}

Theoremrnxrnres 35516* Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021.)
ran (𝑅 ⋉ (𝑆𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑆𝑦)}

Theoremrnxrncnvepres 35517* Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}

Theoremrnxrnidres 35518* Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢 = 𝑦𝑢𝑅𝑥)}

Theoremxrnres 35519 Two ways to express restriction of range Cartesian product, see also xrnres2 35520, xrnres3 35521. (Contributed by Peter Mazsa, 5-Jun-2021.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ 𝑆)

Theoremxrnres2 35520 Two ways to express restriction of range Cartesian product, see also xrnres 35519, xrnres3 35521. (Contributed by Peter Mazsa, 6-Sep-2021.)
((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))

Theoremxrnres3 35521 Two ways to express restriction of range Cartesian product, see also xrnres 35519, xrnres2 35520. (Contributed by Peter Mazsa, 28-Mar-2020.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Theoremxrnres4 35522 Two ways to express restriction of range Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝑆) ∩ (𝐴 × (ran (𝑅𝐴) × ran (𝑆𝐴))))

Theoremxrnresex 35523 Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.)
((𝐴𝑉𝑅𝑊 ∧ (𝑆𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆𝐴)) ∈ V)

Theoremxrnidresex 35524 Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)

Theoremxrncnvepresex 35525 Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)

Theorembrin2 35526 Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))

Theorembrin3 35527 Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆){{𝐵}}))

20.22.4  Cosets by ` R `

Definitiondf-coss 35528* Define the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑢𝑅𝑥 and 𝑢𝑅𝑦 hold, i.e., both 𝑥 and 𝑦 are are elements of the 𝑅 -coset of 𝑢 (see dfcoss2 35530 and the comment of dfec2 8285). 𝑅 is usually a relation.

This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to 𝑅, the right side relate to 𝑅 (see e.g. ~? pet ). Without the definition of 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 35531) or to the range of a range Cartesian product of classes (cf. dfcoss4 35532), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 35530. Technically, we can define it via composition (dfcoss3 35531) or as the range of a range Cartesian product (dfcoss4 35532), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 35783, df-funALTV 35784) and disjoints (dfdisjs 35810, dfdisjs2 35811, df-disjALTV 35807, dfdisjALTV2 35816) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)

𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}

Definitiondf-coels 35529 Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 35544. Possible definitions are the special cases of dfcoss3 35531 and dfcoss4 35532. (Contributed by Peter Mazsa, 20-Nov-2019.)
𝐴 = ≀ ( E ↾ 𝐴)

Theoremdfcoss2 35530* Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8285). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)}

Theoremdfcoss3 35531 Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35528). (Contributed by Peter Mazsa, 27-Dec-2018.)
𝑅 = (𝑅𝑅)

Theoremdfcoss4 35532 Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35528). (Contributed by Peter Mazsa, 12-Jul-2021.)
𝑅 = ran (𝑅𝑅)

Theoremcossex 35533 If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.)
(𝐴𝑉 → ≀ 𝐴 ∈ V)

Theoremcosscnvex 35534 If 𝐴 is a set then the class of cosets by the converse of 𝐴 is a set. (Contributed by Peter Mazsa, 18-Oct-2019.)
(𝐴𝑉 → ≀ 𝐴 ∈ V)

Theorem1cosscnvepresex 35535 Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.)
(𝐴𝑉 → ≀ ( E ↾ 𝐴) ∈ V)

Theorem1cossxrncnvepresex 35536 Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.)
((𝐴𝑉𝑅𝑊) → ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)

Theoremrelcoss 35537 Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
Rel ≀ 𝑅

Theoremrelcoels 35538 Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
Rel ∼ 𝐴

Theoremcossss 35539 Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
(𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Theoremcosseq 35540 Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
(𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Theoremcosseqi 35541 Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.)
𝐴 = 𝐵       𝐴 = ≀ 𝐵

Theoremcosseqd 35542 Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Theorem1cossres 35543* The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
≀ (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑅𝑦)}

Theoremdfcoels 35544* Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}

Theorembrcoss 35545* 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))

Theorembrcoss2 35546* Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅)))

Theorembrcoss3 35547 Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))

Theorembrcosscnvcoss 35548 For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theorembrcoels 35549* 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))

Theoremcocossss 35550* Two ways of saying that cosets by cosets by 𝑅 is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.)
( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))

Theoremcnvcosseq 35551 The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
𝑅 = ≀ 𝑅

Theorembr2coss 35552 Cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))

Theorembr1cossres 35553* 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑅𝐶)))

Theorembr1cossres2 35554* 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑥𝐴 (𝐵 ∈ [𝑥]𝑅𝐶 ∈ [𝑥]𝑅)))

Theoremrelbrcoss 35555* 𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.)
((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))))

Theorembr1cossinres 35556* 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))

Theorembr1cossxrnres 35557* 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))

Theorembr1cossinidres 35558* 𝐵 and 𝐶 are cosets by an intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶))))

Theorembr1cossincnvepres 35559* 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶))))

Theorembr1cossxrnidres 35560* 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by a range Cartesian product with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 = 𝐶𝑢𝑅𝐵) ∧ (𝑢 = 𝐸𝑢𝑅𝐷))))

Theorembr1cossxrncnvepres 35561* 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by a range Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))

Theoremdmcoss3 35562 The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
dom ≀ 𝑅 = dom 𝑅

Theoremdmcoss2 35563 The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
dom ≀ 𝑅 = ran 𝑅

Theoremrncossdmcoss 35564 The range of cosets is the domain of them (this should be rncoss 5841 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
ran ≀ 𝑅 = dom ≀ 𝑅

Theoremdm1cosscnvepres 35565 The domain of cosets of the restricted converse epsilon relation is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019.) (Revised by Peter Mazsa, 26-Sep-2021.)
dom ≀ ( E ↾ 𝐴) = 𝐴

Theoremdmcoels 35566 The domain of coelements in 𝐴 is the union of 𝐴. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Peter Mazsa, 5-Apr-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)
dom ∼ 𝐴 = 𝐴

Theoremeldmcoss 35567* Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
(𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Theoremeldmcoss2 35568 Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
(𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅𝐴𝑅𝐴))

Theoremeldm1cossres 35569* Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
(𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑢𝑅𝐵))

Theoremeldm1cossres2 35570* Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
(𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))

Theoremrefrelcosslem 35571 Lemma for the left side of the refrelcoss3 35572 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥

Theoremrefrelcoss3 35572* The class of cosets by 𝑅 is reflexive, see dfrefrel3 35625. (Contributed by Peter Mazsa, 30-Jul-2019.)
(∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)

Theoremrefrelcoss2 35573 The class of cosets by 𝑅 is reflexive, see dfrefrel2 35624. (Contributed by Peter Mazsa, 30-Jul-2019.)
(( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Theoremsymrelcoss3 35574 The class of cosets by 𝑅 is symmetric, see dfsymrel3 35655. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
(∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)

Theoremsymrelcoss2 35575 The class of cosets by 𝑅 is symmetric, see dfsymrel2 35654. (Contributed by Peter Mazsa, 27-Dec-2018.)
(𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Theoremcossssid 35576 Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 27-Jul-2021.)
( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))

Theoremcossssid2 35577* Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.)
( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))

Theoremcossssid3 35578* Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.)
( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))

Theoremcossssid4 35579* Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.)
( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)

Theoremcossssid5 35580* Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.)
( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))

Theorembrcosscnv 35581* 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))

Theorembrcosscnv2 35582 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 12-Mar-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))

Theorembr1cosscnvxrn 35583 𝐴 and 𝐵 are cosets by the converse range Cartesian product: a binary relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))

Theorem1cosscnvxrn 35584 Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
(𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)

Theoremcosscnvssid3 35585* Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 28-Jul-2021.)
( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))

Theoremcosscnvssid4 35586* Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.)
( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)

Theoremcosscnvssid5 35587* Equivalent expressions for the class of cosets by the converse of the relation 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.)
(( ≀ 𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))

Theoremcoss0 35588 Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
≀ ∅ = ∅

Theoremcossid 35589 Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
≀ I = I

Theoremcosscnvid 35590 Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
I = I

Theoremtrcoss 35591* Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.)
(∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremeleccossin 35592 Two ways of saying that the coset of 𝐴 and the coset of 𝐶 have the common element 𝐵. (Contributed by Peter Mazsa, 15-Oct-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))

Theoremtrcoss2 35593* Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.)
(∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))

20.22.5  Relations

Definitiondf-rels 35594 Define the relations class. Proper class relations (like I, see reli 5696) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 35596).

The class of relations is a great tool we can use when we define classes of different relations as nullary class constants as required by the 2. point in our Guidelines https://us.metamath.org/mpeuni/mathbox.html 35596. When we want to define a specific class of relations as a nullary class constant, the appropriate method is the following:

1. We define the specific nullary class constant for general sets (see e.g. df-refs 35619), then

2. we get the required class of relations by the intersection of the class of general sets above with the class of relations df-rels 35594 (see df-refrels 35620 and the resulting dfrefrels2 35622 and dfrefrels3 35623).

3. Finally, in order to be able to work with proper classes (like iprc 7609) as well, we define the predicate of the relation (see df-refrel 35621) so that it is true for the relevant proper classes (see refrelid 35630), and that the element of the class of the required relations (e.g. elrefrels3 35627) and this predicate are the same in case of sets (see elrefrelsrel 35628). (Contributed by Peter Mazsa, 13-Jun-2018.)

Rels = 𝒫 (V × V)

Theoremelrels2 35595 The element of the relations class (df-rels 35594) and the relation predicate (df-rel 5560) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Theoremelrelsrel 35596 The element of the relations class (df-rels 35594) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Theoremelrelsrelim 35597 The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.)
(𝑅 ∈ Rels → Rel 𝑅)

Theoremelrels5 35598 Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅))

Theoremelrels6 35599 Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅))

Theoremelrelscnveq3 35600* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

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