HomeHome Metamath Proof Explorer
Theorem List (p. 344 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

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))}
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >