P. 391 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sucmapleftuniq 39001	Left uniqueness of the successor mapping. (Contributed by Peter Mazsa, 8-Jan-2026.)
⊢ ((𝐿 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊 ∧ 𝑁 ∈ 𝑋) → ((𝐿 SucMap 𝑁 ∧ 𝑀 SucMap 𝑁) → 𝐿 = 𝑀))
Theorem	exeupre 39002*	Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (∃𝑚 𝑚 SucMap 𝑁 ↔ ∃!𝑚 𝑚 SucMap 𝑁))
Theorem	preex 39003	The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ pre 𝑁 ∈ V
Theorem	eupre2 39004*	Unique predecessor exists on the range of the successor map. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
Theorem	eupre 39005*	Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))
Theorem	presucmap 39006	pre is really a predecessor (when it should be). This correctness theorem for pre makes it usable in proofs without unfolding ℩. This theorem gives one witness; preuniqval 39007 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
Theorem	preuniqval 39007*	Uniqueness/canonicity of pre. presucmap 39006 gives one witness; this theorem gives it is the only one. It turns any predecessor proof into an equality with pre 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))
Theorem	sucpre 39008	suc is a right-inverse of pre on Suc. This theorem states the partial inverse relation in the direction we most often need. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ Suc → suc pre 𝑁 = 𝑁)
Theorem	presuc 39009	pre is a left-inverse of suc. This theorem gives a clean rewrite rule that eliminates pre on explicit successors. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑀 ∈ 𝑉 → pre suc 𝑀 = 𝑀)
Theorem	press 39010	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ⊆ 𝑁)
Theorem	preel 39011	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ∈ 𝑁)
21.26.7  Cosets by ` R `
Definition	df-coss 39012*	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 39014 and the comment of dfec2 8685). 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 39476). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 39015) or to the range of a range Cartesian product of classes (cf. dfcoss4 39016), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 39014. Technically, we can define it via composition (dfcoss3 39015) or as the range of a range Cartesian product (dfcoss4 39016), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 39277, df-funALTV 39278) and disjoints (dfdisjs 39304, dfdisjs2 39305, df-disjALTV 39301, dfdisjALTV2 39310) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Definition	df-coels 39013	Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 39031. Possible definitions are the special cases of dfcoss3 39015 and dfcoss4 39016. (Contributed by Peter Mazsa, 20-Nov-2019.)
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
Theorem	dfcoss2 39014*	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 8685). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}
Theorem	dfcoss3 39015	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 39012). (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
Theorem	dfcoss4 39016	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 39012). (Contributed by Peter Mazsa, 12-Jul-2021.)
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)
Theorem	cosscnv 39017*	Class of cosets by the converse of 𝑅. (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
Theorem	coss1cnvres 39018*	Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}
Theorem	coss2cnvepres 39019*	Special case of coss1cnvres 39018. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(◡ E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))}
Theorem	cossex 39020	If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V)
Theorem	cosscnvex 39021	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.)
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ V)
Theorem	1cosscnvepresex 39022	Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V)
Theorem	1cossxrncnvepresex 39023	Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V)
Theorem	relcoss 39024	Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ Rel ≀ 𝑅
Theorem	relcoels 39025	Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ Rel ∼ 𝐴
Theorem	cossss 39026	Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Theorem	cosseq 39027	Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Theorem	cosseqi 39028	Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ≀ 𝐴 = ≀ 𝐵
Theorem	cosseqd 39029	Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)
Theorem	1cossres 39030*	The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Theorem	dfcoels 39031*	Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
Theorem	brcoss 39032*	𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
Theorem	brcoss2 39033*	Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅)))
Theorem	brcoss3 39034	Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅))
Theorem	brcosscnvcoss 39035	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴))
Theorem	brcoels 39036*	𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢)))
Theorem	cocossss 39037*	Two ways of saying that cosets by cosets by 𝑅 is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.)
⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
Theorem	cnvcosseq 39038	The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ◡ ≀ 𝑅 = ≀ 𝑅
Theorem	br2coss 39039	Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Theorem	br1cossres 39040*	𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑅𝐶)))
Theorem	br1cossres2 39041*	𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅)))
Theorem	brressn 39042	Binary relation on a restriction to a singleton. (Contributed by Peter Mazsa, 11-Jun-2024.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)))
Theorem	ressn2 39043*	A class ' R ' restricted to the singleton of the class ' A ' is the ordered pair class abstraction of the class ' A ' and the sets in relation ' R ' to ' A ' (and not in relation to the singleton ' { A } ' ). (Contributed by Peter Mazsa, 16-Jun-2024.)
⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}
Theorem	refressn 39044*	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 39043) is reflexive, see also refrelressn 39115. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Theorem	antisymressn 39045	Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 39043) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)
⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Theorem	trressn 39046	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 39043) is transitive, see also trrelressn 39178. (Contributed by Peter Mazsa, 16-Jun-2024.)
⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Theorem	relbrcoss 39047*	𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅))))
Theorem	br1cossinres 39048*	𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))
Theorem	br1cossxrnres 39049*	⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by an range Cartesian product with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))
Theorem	br1cossinidres 39050*	𝐵 and 𝐶 are cosets by an intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))))
Theorem	br1cossincnvepres 39051*	𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))))
Theorem	br1cossxrnidres 39052*	⟨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.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))))
Theorem	br1cossxrncnvepres 39053*	⟨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.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))))
Theorem	dmcoss3 39054	The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ dom ≀ 𝑅 = dom ◡𝑅
Theorem	dmcoss2 39055	The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ dom ≀ 𝑅 = ran 𝑅
Theorem	rncossdmcoss 39056	The range of cosets is the domain of them (this should be rncoss 5958 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
⊢ ran ≀ 𝑅 = dom ≀ 𝑅
Theorem	dm1cosscnvepres 39057	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
Theorem	dmcoels 39058	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
Theorem	eldmcoss 39059*	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	eldmcoss2 39060	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴))
Theorem	eldm1cossres 39061*	Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑢𝑅𝐵))
Theorem	eldm1cossres2 39062*	Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅))
Theorem	refrelcosslem 39063	Lemma for the left side of the refrelcoss3 39064 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
Theorem	refrelcoss3 39064*	The class of cosets by 𝑅 is reflexive, see dfrefrel3 39107. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)
Theorem	refrelcoss2 39065	The class of cosets by 𝑅 is reflexive, see dfrefrel2 39106. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	symrelcoss3 39066	The class of cosets by 𝑅 is symmetric, see dfsymrel3 39145. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)
Theorem	symrelcoss2 39067	The class of cosets by 𝑅 is symmetric, see dfsymrel2 39144. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	cossssid 39068	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)))
Theorem	cossssid2 39069*	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 ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
Theorem	cossssid3 39070*	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 ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
Theorem	cossssid4 39071*	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)
Theorem	cossssid5 39072*	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 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅))
Theorem	brcosscnv 39073*	𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
Theorem	brcosscnv2 39074	𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 12-Mar-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
Theorem	br1cosscnvxrn 39075	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ (𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵)))
Theorem	1cosscnvxrn 39076	Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵)
Theorem	cosscnvssid3 39077*	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 ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
Theorem	cosscnvssid4 39078*	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)
Theorem	cosscnvssid5 39079*	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 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
Theorem	coss0 39080	Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
⊢ ≀ ∅ = ∅
Theorem	cossid 39081	Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
⊢ ≀ I = I
Theorem	cosscnvid 39082	Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ≀ ◡ I = I
Theorem	trcoss 39083*	Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.)
⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
Theorem	eleccossin 39084	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.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶)))
Theorem	trcoss2 39085*	Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.)
⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
Theorem	cosselrels 39086	Cosets of sets are elements of the relations class. Implies ⊢ (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.)
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ Rels )
Theorem	cnvelrels 39087	The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )
Theorem	cosscnvelrels 39088	Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels )
21.26.8  Subset relations
Definition	df-ssr 39089*	Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5552) and identity relation (df-id 5547) classes. Subset relation class and Scott Fenton's subset class df-sset 36217 are the same: S = SSet (compare dfssr2 39090 with df-sset 36217), the only reason we do not use dfssr2 39090 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 subsets and the subclass relationship (df-ss 3924) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 39092. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3924. The only exception (aside from directly investigating the class S e.g. in relssr 39091 or in extssr 39100) is when we have a specific purpose with its usage, like in case of df-refs 39101 versus df-cnvrefs 39116, 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 38827, extep 38800 and extssr 39100, 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⟩ ∣ 𝑥 ⊆ 𝑦}
Theorem	dfssr2 39090	Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Theorem	relssr 39091	The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ Rel S
Theorem	brssr 39092	The subset relation and subclass relationship (df-ss 3924) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	brssrid 39093	Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴)
Theorem	issetssr 39094	Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Theorem	brssrres 39095	Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))
Theorem	br1cnvssrres 39096	Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))
Theorem	brcnvssr 39097	The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴))
Theorem	brcnvssrid 39098	Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴)
Theorem	br1cossxrncnvssrres 39099*	⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by range Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷))))
Theorem	extssr 39100	Property of subset relation, see also extid 38827, extep 38800 and the comment of df-ssr 39089. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵))