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	brsucmap 38801	Binary relation form of the successor map, general version. (Contributed by Peter Mazsa, 6-Jan-2026.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))
Theorem	relsucmap 38802	The successor map is a relation. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ Rel SucMap
Theorem	dmsucmap 38803	The domain of the successor map is the universe. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ dom SucMap = V
Definition	df-succl 38804	Define Suc as the class of all successors, i.e. the range of the successor map: 𝑛 ∈ Suc iff ∃𝑚suc 𝑚 = 𝑛 (see dfsuccl2 38805). By injectivity of suc (suc11reg 9531), every 𝑛 ∈ Suc has at most one predecessor, which is exactly what pre 𝑛 (df-pre 38810) names. Cf. dfsuccl3 38808 and dfsuccl4 38809. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ Suc = ran SucMap
Theorem	dfsuccl2 38805*	Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 29-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
Theorem	mopre 38806*	There is at most one predecessor of 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ ∃*𝑚 suc 𝑚 = 𝑁
Theorem	exeupre2 38807*	Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁)
Theorem	dfsuccl3 38808*	Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 30-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
Theorem	dfsuccl4 38809*	Alternate definition that incorporates the most desirable properties of the successor class. (Contributed by Peter Mazsa, 30-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}
Definition	df-pre 38810*	Define the term-level successor-predecessor. It is the unique 𝑚 with suc 𝑚 = 𝑁 when such an 𝑚 exists; otherwise pre 𝑁 is the arbitrary default chosen by ℩. See its alternate definitions dfpre 38811, dfpre2 38812, dfpre3 38813 and dfpre4 38815. Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6259 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6271, cf. also df-bnj14 34848) in several respects. Its most abstract property as a specialisation is that it has a unique existing value by default. This is in contrast to the general version. The uniqueness (conditional on existence) is implied by the property of this specific instance of the general case involving the successor map df-sucmap 38797 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38825, which originates from suc11reg 9531. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5840. Note that dom SucMap = V (see dmsucmap 38803), so the equivalent definition dfpre 38811 uses (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁)). (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
Theorem	dfpre 38811*	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁))
Theorem	dfpre2 38812*	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
Theorem	dfpre3 38813*	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))
Theorem	dfpred4 38814	Alternate definition of the predecessor class when 𝑁 is a set. (Contributed by Peter Mazsa, 26-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]◡(𝑅 ↾ 𝐴))
Theorem	dfpre4 38815*	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 38797). (Contributed by Peter Mazsa, 26-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
Definition	df-blockliftfix 38816*	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 38794), 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 8643 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 38817	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 5633. 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 38818	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 38819	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 38820	Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
Theorem	sucdifsn 38821	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 38822	The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38823. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	ressucdifsn 38823	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 38824	A set is succeeded by its successor. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)
Theorem	sucmapleftuniq 38825	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 38826*	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 38827	The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ pre 𝑁 ∈ V
Theorem	eupre2 38828*	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 38829*	Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))
Theorem	presucmap 38830	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 38831 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
Theorem	preuniqval 38831*	Uniqueness/canonicity of pre. presucmap 38830 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 38832	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 38833	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 38834	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ⊆ 𝑁)
Theorem	preel 38835	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 38836*	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 38838 and the comment of dfec2 8639). 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 39300). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 38839) or to the range of a range Cartesian product of classes (cf. dfcoss4 38840), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 38838. Technically, we can define it via composition (dfcoss3 38839) or as the range of a range Cartesian product (dfcoss4 38840), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 39101, df-funALTV 39102) and disjoints (dfdisjs 39128, dfdisjs2 39129, df-disjALTV 39125, dfdisjALTV2 39134) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Definition	df-coels 38837	Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 38855. Possible definitions are the special cases of dfcoss3 38839 and dfcoss4 38840. (Contributed by Peter Mazsa, 20-Nov-2019.)
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
Theorem	dfcoss2 38838*	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 8639). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}
Theorem	dfcoss3 38839	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38836). (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
Theorem	dfcoss4 38840	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38836). (Contributed by Peter Mazsa, 12-Jul-2021.)
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)
Theorem	cosscnv 38841*	Class of cosets by the converse of 𝑅. (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
Theorem	coss1cnvres 38842*	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 38843*	Special case of coss1cnvres 38842. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(◡ E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))}
Theorem	cossex 38844	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 38845	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 38846	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 38847	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 38848	Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ Rel ≀ 𝑅
Theorem	relcoels 38849	Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ Rel ∼ 𝐴
Theorem	cossss 38850	Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Theorem	cosseq 38851	Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Theorem	cosseqi 38852	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 38853	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 38854*	The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Theorem	dfcoels 38855*	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 38856*	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 38857*	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 38858	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 38859	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 38860*	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 38861*	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 38862	The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ◡ ≀ 𝑅 = ≀ 𝑅
Theorem	br2coss 38863	Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Theorem	br1cossres 38864*	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 38865*	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 38866	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 38867*	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 38868*	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38867) is reflexive, see also refrelressn 38939. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Theorem	antisymressn 38869	Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38867) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)
⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Theorem	trressn 38870	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38867) is transitive, see also trrelressn 39002. (Contributed by Peter Mazsa, 16-Jun-2024.)
⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Theorem	relbrcoss 38871*	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 38872*	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 38873*	⟨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 38874*	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 38875*	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 38876*	⟨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 38877*	⟨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 38878	The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ dom ≀ 𝑅 = dom ◡𝑅
Theorem	dmcoss2 38879	The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ dom ≀ 𝑅 = ran 𝑅
Theorem	rncossdmcoss 38880	The range of cosets is the domain of them (this should be rncoss 5926 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
⊢ ran ≀ 𝑅 = dom ≀ 𝑅
Theorem	dm1cosscnvepres 38881	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 38882	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 38883*	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	eldmcoss2 38884	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴))
Theorem	eldm1cossres 38885*	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 38886*	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 38887	Lemma for the left side of the refrelcoss3 38888 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
Theorem	refrelcoss3 38888*	The class of cosets by 𝑅 is reflexive, see dfrefrel3 38931. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)
Theorem	refrelcoss2 38889	The class of cosets by 𝑅 is reflexive, see dfrefrel2 38930. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	symrelcoss3 38890	The class of cosets by 𝑅 is symmetric, see dfsymrel3 38969. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)
Theorem	symrelcoss2 38891	The class of cosets by 𝑅 is symmetric, see dfsymrel2 38968. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	cossssid 38892	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 38893*	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 38894*	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 38895*	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 38896*	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 38897*	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 38898	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 38899	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 38900	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)