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Theorem List for Metamath Proof Explorer - 34301-34400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrnres2 34301 Two ways to express restriction of range Cartesian product, cf. xrnres 34300, xrnres3 34302. (Contributed by Peter Mazsa, 6-Sep-2021.)
((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))

Theoremxrnres3 34302 Two ways to express restriction of range Cartesian product, cf. xrnres 34300, xrnres2 34301. (Contributed by Peter Mazsa, 28-Mar-2020.)
((𝑅𝑆) ↾ 𝐴) = ((𝑅𝐴) ⋉ (𝑆𝐴))

Theoremxrnres4 34303 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 34304 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 34305 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 34306 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 34307 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 34308 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}}))

20.21.4  Cosets by ` R `

Definitiondf-coss 34309* 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 (cf. dfcoss2 34311 and the comment of dfec2 7790). 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 (cf. 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 34312) or to the range of a range Cartesian product of classes (cf. dfcoss4 34313), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 34311. Technically, we can define it via composition (dfcoss3 34312) or as the range of a range Cartesian product (dfcoss4 34313), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions ( ~? df-funsALTV , ~? df-funALTV ) and disjoints ( ~? dfdisjs , ~? dfdisjs2 , ~? df-disjALTV , ~? dfdisjALTV2 ) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)

𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}

Definitiondf-coels 34310 Define the class of coelements on the class 𝐴, cf. the alternate definition dfcoels 34325. Possible definitions are the special cases of dfcoss3 34312 and dfcoss4 34313. (Contributed by Peter Mazsa, 20-Nov-2019.)
𝐴 = ≀ ( E ↾ 𝐴)

Theoremdfcoss2 34311* 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 (cf. the comment of dfec2 7790). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)}

Theoremdfcoss3 34312 Alternate definition of the class of cosets by 𝑅 (cf. the comment of df-coss 34309). (Contributed by Peter Mazsa, 27-Dec-2018.)
𝑅 = (𝑅𝑅)

Theoremdfcoss4 34313 Alternate definition of the class of cosets by 𝑅 (cf. the comment of df-coss 34309). (Contributed by Peter Mazsa, 12-Jul-2021.)
𝑅 = ran (𝑅𝑅)

Theoremcossex 34314 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 34315 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 34316 Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.)
(𝐴𝑉 → ≀ ( E ↾ 𝐴) ∈ V)

Theorem1cossxrncnvepresex 34317 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 34318 Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
Rel ≀ 𝑅

Theoremrelcoels 34319 Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
Rel ∼ 𝐴

Theoremcossss 34320 Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
(𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Theoremcosseq 34321 Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
(𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Theoremcosseqi 34322 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 34323 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 34324* The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
≀ (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑅𝑦)}

Theoremdfcoels 34325* Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}

Theorembrcoss 34326* 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))

Theorembrcoss2 34327* 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 34328 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 34329 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 34330* 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 34331* 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 34332 The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
𝑅 = ≀ 𝑅

Theorembr2coss 34333 Cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))

Theorembr1cossres 34334* 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 34335* 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 34336* 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 34337* 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 34338* 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 34339* 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 34340* 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 34341* 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 34342* 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 34343 The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
dom ≀ 𝑅 = dom 𝑅

Theoremdmcoss2 34344 The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
dom ≀ 𝑅 = ran 𝑅

Theoremrncossdmcoss 34345 The range of cosets is the domain of them (this should be rncoss 5418 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
ran ≀ 𝑅 = dom ≀ 𝑅

Theoremdm1cosscnvepres 34346 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 34347 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 34348* Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
(𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Theoremeldmcoss2 34349 Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
(𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅𝐴𝑅𝐴))

Theoremeldm1cossres 34350* Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
(𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑢𝑅𝐵))

Theoremeldm1cossres2 34351* Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
(𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))

Theoremrefrelcosslem 34352 Lemma for the left side of the refrelcoss3 34353 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥

Theoremrefrelcoss3 34353* The class of cosets by 𝑅 is reflexive, cf. dfrefrel3 34406. (Contributed by Peter Mazsa, 30-Jul-2019.)
(∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)

Theoremrefrelcoss2 34354 The class of cosets by 𝑅 is reflexive, cf. dfrefrel2 34405. (Contributed by Peter Mazsa, 30-Jul-2019.)
(( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Theoremsymrelcoss3 34355 The class of cosets by 𝑅 is symmetric, cf. dfsymrel3 34436. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
(∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)

Theoremsymrelcoss2 34356 The class of cosets by 𝑅 is symmetric, cf. dfsymrel2 34435. (Contributed by Peter Mazsa, 27-Dec-2018.)
(𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Theoremcossssid 34357 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 34358* 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 34359* 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 34360* 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 34361* 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 34362* 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))

Theorembrcosscnv2 34363 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 34364 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 34365 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 34366* 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 34367* 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 34368* 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 34369 Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
≀ ∅ = ∅

Theoremcossid 34370 Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
≀ I = I

Theoremcosscnvid 34371 Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
I = I

Theoremtrcoss 34372* Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.)
(∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremeleccossin 34373 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 34374* 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) ≠ ∅))

20.21.5  Relations

Definitiondf-rels 34375 Define the relations class. Proper class relations (like I, cf. reli 5282) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (cf. elrelsrel 34377).

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 http://us.metamath.org/mpeuni/mathbox.html. 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 (cf. e.g. df-refs 34400), 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 34375 (cf. df-refrels 34401 and the resulting dfrefrels2 34403 and dfrefrels3 34404).

3. Finally, in order to be able to work with proper classes (like iprc 7143) as well, we define the predicate of the relation (cf. df-refrel 34402) so that it is true for the relevant proper classes (cf. refrelid 34411), and that the element of the class of the required relations (e.g. elrefrels3 34408) and this predicate are the same in case of sets (cf. elrefrelsrel 34409). (Contributed by Peter Mazsa, 13-Jun-2018.)

Rels = 𝒫 (V × V)

Theoremelrels2 34376 The element of the relations class (df-rels 34375) and the relation predicate (df-rel 5150) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Theoremelrelsrel 34377 The element of the relations class (df-rels 34375) 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 34378 The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.)
(𝑅 ∈ Rels → Rel 𝑅)

Theoremelrels5 34379 Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅))

Theoremelrels6 34380 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 34381* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremelrelscnveq 34382 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))

Theoremelrelscnveq2 34383* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremelrelscnveq4 34384* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremcnvelrels 34385 The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.)
(𝐴𝑉𝐴 ∈ Rels )

Theoremcosselrels 34386 Cosets of sets are elements of the relations class. Implies (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.)
(𝐴𝑉 → ≀ 𝐴 ∈ Rels )

Theoremcosscnvelrels 34387 Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐴𝑉 → ≀ 𝐴 ∈ Rels )

20.21.6  Subset relations

Definitiondf-ssr 34388* Define the subsets class or the class of all subset relations. Similar to definitions of epsilon relation (df-eprel 5058) and identity relation (df-id 5053) classes. Subset relation class and Scott Fenton's subset class df-sset 32088 are the same: S = SSet (compare dfssr2 34389 with df-sset 32088, cf. comment of df-xrn 34273), the only reason we do not use dfssr2 34389 as the base definition of the subsets class is the way we defined the epsilon relation and the identity relation classes.

The binary relation on the class of all subsets and the subclass relationship (df-ss 3621) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set, cf. brssr 34391. Yet in general we use the subclass relation 𝐴𝐵 both for classes and for sets, cf. the comment of df-ss 3621. The only exception (aside from directly investigating the class S e.g. in relssr 34390 or in extssr 34399) is when we have a specific purpose with its usage, like in case of df-refs 34400 vs. df-cnvrefs 34413, where we need S to define the class of reflexive sets in order to be able to define the class of converse reflexive sets with the help of the converse of S.

The subsets class S has another place in set.mm as well: if we define extensional relation based on the common property in extid 34222, extep 34189 and extssr 34399, then "extrelssr" " |- ExtRel _S " is a theorem along with "extrelep" " |- ExtRel _E " and "extrelid" " |- ExtRel _I ". (Contributed by Peter Mazsa, 25-Jul-2019.)

S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}

Theoremdfssr2 34389 Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))

Theoremrelssr 34390 The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
Rel S

Theorembrssr 34391 The subset relation and subclass relationship (df-ss 3621) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.)
(𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))

Theorembrssrid 34392 Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴𝑉𝐴 S 𝐴)

Theoremissetssr 34393 Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴 ∈ V ↔ 𝐴 S 𝐴)

Theorembrssrres 34394 Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
(𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵𝐶)))

Theorembr1cnvssrres 34395 Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
(𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))

Theorembrcnvssr 34396 The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))

Theorembrcnvssrid 34397 Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
(𝐴𝑉𝐴 S 𝐴)

Theorembr1cossxrncnvssrres 34398* 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by tail Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))

Theoremextssr 34399 Property of subset relation, cf. extid 34222, extep 34189 and the comment of df-ssr 34388. (Contributed by Peter Mazsa, 10-Jul-2019.)
((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))

20.21.7  Reflexivity

Definitiondf-refs 34400 Define the class of all reflexive sets. It is used only by df-refrels 34401. We use subset relation S (df-ssr 34388) here to be able to define converse reflexivity (df-cnvrefs 34413), cf. the comment of df-ssr 34388. The elements of this class are not necessarily relations (vs. df-refrels 34401).

Note the similarity of the definitions df-refs 34400, df-syms 34428 and ~? df-trs , cf. the comments of dfrefrels2 34403. (Contributed by Peter Mazsa, 19-Jul-2019.)

Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

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