P. 389 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfpre4 38801*	Alternate definition of the predecessor of the 𝑁 set. The ◡ SucMap is just the "PreMap"; we did not define it because we do not expect to use it extensively in future (cf. the comments of df-sucmap 38783). (Contributed by Peter Mazsa, 26-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
Definition	df-blockliftfix 38802*	Define the equilibrium / fixed-point condition for "block carriers". Start with a candidate block-family 𝑎 (a set whose elements you intend to treat as blocks). Combine it with a relation 𝑟 by forming the block-lift span 𝑇 = (𝑟 ⋉ (◡ E ↾ 𝑎)). For a block 𝑢 ∈ 𝑎, the fiber [𝑢]𝑇 is the set of all outputs produced from "external targets" of 𝑟 together with "internal members" of 𝑢; in other words, 𝑇 is the mechanism that generates new blocks from old ones. Now apply the standard quotient construction (dom 𝑇 / 𝑇). This produces the family of all T-blocks (the cosets [𝑥]𝑇 of witnesses 𝑥 in the domain of 𝑇). In general, this operation can change your carrier: starting from 𝑎, it may generate a different block-family (dom 𝑇 / 𝑇). The equation (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎 says exactly: if you generate blocks from 𝑎 using the lift determined by 𝑟 (cf. df-blockliftmap 38780), you get back the same 𝑎. So 𝑎 is stable under the block-generation operator induced by 𝑟. This is why it is a genuine fixpoint/equilibrium condition: one application of the "make-the-blocks" operator causes no carrier drift, i.e. no hidden refinement/coarsening of what counts as a block. Here, the quotient (dom 𝑇 / 𝑇) is the standard carrier of 𝑇 -blocks; see dfqs2 8650 for the quotient-as-range viewpoint. This is an untyped equilibrium predicate on pairs ⟨𝑟, 𝑎⟩. No hypothesis 𝑟 ∈ Rels is built into the definition, because the fixpoint equation depends only on those ordered pairs ⟨𝑥, 𝑦⟩ that belong to 𝑟 and hence can witness an atomic instance 𝑥𝑟𝑦; extra non-ordered-pair "junk" elements in 𝑟 are ignored automatically by the relational membership predicate. When later work needs 𝑟 to be relation-typed (e.g. to intersect with ( Rels × V)-style typedness modules, or to apply Rels-based infrastructure uniformly), the additional typing constraint 𝑟 ∈ Rels should be imposed locally as a separate conjunct (rather than being baked into this equilibrium module). (Contributed by Peter Mazsa, 25-Jan-2026.) (Revised by Peter Mazsa, 20-Feb-2026.)
⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎}
Definition	df-shiftstable 38803	Define shift-stability, a general "procedure" pattern for "the one-step backward shift/transport of 𝐹 along 𝑆", and then ∩ 𝐹 enforces "and it already holds here". Let 𝐹 be a relation encoding a property that depends on a "level" coordinate (for example, a feasibility condition indexed by a carrier, a grade, or a stage in a construction). Let 𝑆 be a shift relation between levels (for example, the successor map SucMap, or any other grading step). The composed relation (𝑆 ∘ 𝐹) transports 𝐹 one step along the shift: 𝑟(𝑆 ∘ 𝐹)𝑛 means there exists a predecessor level 𝑚 such that 𝑟𝐹𝑚 and 𝑚𝑆𝑛 (e.g., 𝑚 SucMap 𝑛). We do not introduce a separate notation for "Shift" because it is simply the standard relational composition df-co 5640. The intersection ((𝑆 ∘ 𝐹) ∩ 𝐹) is the locally shift-stable fragment of 𝐹: it consists exactly of those points where the property holds at some immediate predecessor that shifts to 𝑛 and also holds at level 𝑛. In other words, it isolates the part of 𝐹 that is already compatible with one-step tower coherence. This definition packages a common construction pattern used throughout the development: "constrain by one-step stability under a chosen shift, then additionally constrain by 𝐹". Iterating the operator (𝑋 ↦ ((𝑆 ∘ 𝑋) ∩ 𝑋) corresponds to multi-step/tower coherence; the one-step definition here is the economical kernel from which such "tower" readings can be developed when needed. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
Theorem	shiftstableeq2 38804	Equality theorem for shift-stability of two classes. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))
Theorem	suceqsneq 38805	One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))
Theorem	sucdifsn2 38806	Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
Theorem	sucdifsn 38807	The difference between the successor and the singleton of a class is the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴
Theorem	ressucdifsn2 38808	The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38809. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	ressucdifsn 38809	The difference between restrictions to the successor and the singleton of a class is the restriction to the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	sucmapsuc 38810	A set is succeeded by its successor. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)
Theorem	sucmapleftuniq 38811	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 38812*	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 38813	The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ pre 𝑁 ∈ V
Theorem	eupre2 38814*	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 38815*	Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))
Theorem	presucmap 38816	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 38817 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
Theorem	preuniqval 38817*	Uniqueness/canonicity of pre. presucmap 38816 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 38818	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 38819	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 38820	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ⊆ 𝑁)
Theorem	preel 38821	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 38822*	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 38824 and the comment of dfec2 8646). 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 39286). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 38825) or to the range of a range Cartesian product of classes (cf. dfcoss4 38826), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 38824. Technically, we can define it via composition (dfcoss3 38825) or as the range of a range Cartesian product (dfcoss4 38826), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 39087, df-funALTV 39088) and disjoints (dfdisjs 39114, dfdisjs2 39115, df-disjALTV 39111, dfdisjALTV2 39120) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Definition	df-coels 38823	Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 38841. Possible definitions are the special cases of dfcoss3 38825 and dfcoss4 38826. (Contributed by Peter Mazsa, 20-Nov-2019.)
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
Theorem	dfcoss2 38824*	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 8646). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}
Theorem	dfcoss3 38825	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38822). (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
Theorem	dfcoss4 38826	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38822). (Contributed by Peter Mazsa, 12-Jul-2021.)
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)
Theorem	cosscnv 38827*	Class of cosets by the converse of 𝑅. (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
Theorem	coss1cnvres 38828*	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 38829*	Special case of coss1cnvres 38828. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(◡ E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))}
Theorem	cossex 38830	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 38831	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 38832	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 38833	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 38834	Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ Rel ≀ 𝑅
Theorem	relcoels 38835	Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ Rel ∼ 𝐴
Theorem	cossss 38836	Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Theorem	cosseq 38837	Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Theorem	cosseqi 38838	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 38839	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 38840*	The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Theorem	dfcoels 38841*	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 38842*	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 38843*	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 38844	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 38845	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 38846*	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 38847*	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 38848	The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ◡ ≀ 𝑅 = ≀ 𝑅
Theorem	br2coss 38849	Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Theorem	br1cossres 38850*	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 38851*	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 38852	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 38853*	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 38854*	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38853) is reflexive, see also refrelressn 38925. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Theorem	antisymressn 38855	Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38853) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)
⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Theorem	trressn 38856	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38853) is transitive, see also trrelressn 38988. (Contributed by Peter Mazsa, 16-Jun-2024.)
⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Theorem	relbrcoss 38857*	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 38858*	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 38859*	⟨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 38860*	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 38861*	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 38862*	⟨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 38863*	⟨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 38864	The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ dom ≀ 𝑅 = dom ◡𝑅
Theorem	dmcoss2 38865	The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ dom ≀ 𝑅 = ran 𝑅
Theorem	rncossdmcoss 38866	The range of cosets is the domain of them (this should be rncoss 5932 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
⊢ ran ≀ 𝑅 = dom ≀ 𝑅
Theorem	dm1cosscnvepres 38867	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 38868	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 38869*	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	eldmcoss2 38870	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴))
Theorem	eldm1cossres 38871*	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 38872*	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 38873	Lemma for the left side of the refrelcoss3 38874 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
Theorem	refrelcoss3 38874*	The class of cosets by 𝑅 is reflexive, see dfrefrel3 38917. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)
Theorem	refrelcoss2 38875	The class of cosets by 𝑅 is reflexive, see dfrefrel2 38916. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	symrelcoss3 38876	The class of cosets by 𝑅 is symmetric, see dfsymrel3 38955. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)
Theorem	symrelcoss2 38877	The class of cosets by 𝑅 is symmetric, see dfsymrel2 38954. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	cossssid 38878	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 38879*	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 38880*	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 38881*	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 38882*	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 38883*	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 38884	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 38885	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 38886	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 38887*	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 38888*	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 38889*	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 38890	Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
⊢ ≀ ∅ = ∅
Theorem	cossid 38891	Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
⊢ ≀ I = I
Theorem	cosscnvid 38892	Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ≀ ◡ I = I
Theorem	trcoss 38893*	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 38894	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 38895*	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 38896	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 38897	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 38898	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 38899*	Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5531) and identity relation (df-id 5526) classes. Subset relation class and Scott Fenton's subset class df-sset 36036 are the same: S = SSet (compare dfssr2 38900 with df-sset 36036), the only reason we do not use dfssr2 38900 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 3906) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 38902. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3906. The only exception (aside from directly investigating the class S e.g. in relssr 38901 or in extssr 38910) is when we have a specific purpose with its usage, like in case of df-refs 38911 versus df-cnvrefs 38926, 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 38637, extep 38610 and extssr 38910, 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 38900	Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))