P. 388 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfadjliftmap 38701*	Alternate (expanded) definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ◡ E ) ↾ 𝐴))
Theorem	dfadjliftmap2 38702*	Alternate definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))
Theorem	blockadjliftmap 38703*	A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ◡ E ) and then adjoining elements as "BlockAdj". Combined, it uses the relation ((𝑅 ⋉ ◡ E ) ∪ ◡ E ). (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ ((𝑅 ⋉ ◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}
Definition	df-blockliftmap 38704	Define the block lift map. Given a relation 𝑅 and a carrier/set 𝐴, we form the block relation (𝑅 ⋉ ◡ E ) (i.e., "follow both 𝑅 and element"), restricted to 𝐴 (or, equivalently, "follow both 𝑅 and elements-of-A", cf. xrnres2 38671). Then map each domain element 𝑚 to its coset [𝑚] under that restricted block relation. For 𝑚 in the domain, which requires (𝑚 ∈ 𝐴 ∧ 𝑚 ≠ ∅ ∧ [𝑚]𝑅 ≠ ∅) (cf. eldmxrncnvepres 38679), the fiber has the product form [𝑚](𝑅 ⋉ ◡ E ) = ([𝑚]𝑅 × 𝑚), so the block relation lifts a block 𝑚 to the rectangular grid "external labels × internal members", see dfblockliftmap2 38706. Contrast: while the adjoined lift, via (𝑅 ∪ ◡ E ), attaches neighbors and members in a single relation (see dfadjliftmap2 38702), the block lift labels each internal member by each external neighbor. For the general case and a two-stage construction (first block lift, then adjoin membership), see the comments to df-adjliftmap 38700. For the equilibrium condition, see df-blockliftfix 38726. (Contributed by Peter Mazsa, 24-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 BlockLiftMap 𝐴) = QMap (𝑅 ⋉ (◡ E ↾ 𝐴))
Theorem	dfblockliftmap 38705*	Alternate definition of the block lift map. (Contributed by Peter Mazsa, 29-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (◡ E ↾ 𝐴)))
Theorem	dfblockliftmap2 38706*	Alternate definition of the block lift map. (Contributed by Peter Mazsa, 29-Jan-2026.)
⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))
Definition	df-sucmap 38707*	Define the successor map, directly as the graph of the successor operation, using only elementary set theory (ordered-pair class abstraction). This avoids committing to any particular construction of the successor function/class from other operators (e.g. a union/composition presentation), while remaining provably equivalent to those presentations (cf. dfsucmap2 38709 and dfsucmap3 38708 vs. df-succf 36083 and dfsuccf2 36154). For maximum mappy shape, see dfsucmap4 38710. We also treat the successor relation as the default shift relation for grading/tower arguments (cf. df-shiftstable 38727). Because it is used pervasively in shift-lift infrastructure, we adopt the short name SucMap rather than the fully systematic "SucAdjLiftMap". You may also define the predecessor relation as the converse graph "PreMap" as ◡ SucMap, which reverses successor edges ( cf. cnvopab 6102) and sends each successor to its (unique) predecessor when it exists. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
Theorem	dfsucmap3 38708	Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ SucMap = ( I AdjLiftMap V)
Theorem	dfsucmap2 38709	Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ SucMap = ( I AdjLiftMap dom I )
Theorem	dfsucmap4 38710	Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)
Theorem	brsucmap 38711	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 38712	The successor map is a relation. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ Rel SucMap
Theorem	dmsucmap 38713	The domain of the successor map is the universe. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ dom SucMap = V
Definition	df-succl 38714	Define Suc as the class of all successors, i.e. the range of the successor map: 𝑛 ∈ Suc iff ∃𝑚suc 𝑚 = 𝑛 (see dfsuccl2 38715). By injectivity of suc (suc11reg 9540), every 𝑛 ∈ Suc has at most one predecessor, which is exactly what pre 𝑛 (df-pre 38720) names. Cf. dfsuccl3 38718 and dfsuccl4 38719. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ Suc = ran SucMap
Theorem	dfsuccl2 38715*	Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 29-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
Theorem	mopre 38716*	There is at most one predecessor of 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ ∃*𝑚 suc 𝑚 = 𝑁
Theorem	exeupre2 38717*	Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁)
Theorem	dfsuccl3 38718*	Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 30-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
Theorem	dfsuccl4 38719*	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 38720*	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 38721, dfpre2 38722, dfpre3 38723 and dfpre4 38725. Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6267 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6279, cf. also df-bnj14 34865) 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 38707 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38735, which originates from suc11reg 9540. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5848. Note that dom SucMap = V (see dmsucmap 38713), so the equivalent definition dfpre 38721 uses (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁)). (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
Theorem	dfpre 38721*	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁))
Theorem	dfpre2 38722*	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
Theorem	dfpre3 38723*	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))
Theorem	dfpred4 38724	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 38725*	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 38707). (Contributed by Peter Mazsa, 26-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]◡ SucMap ))
Definition	df-blockliftfix 38726*	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 38704), 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 8652 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 38727	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 5641. 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 38728	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 38729	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 38730	Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
Theorem	sucdifsn 38731	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 38732	The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38733. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
Theorem	ressucdifsn 38733	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 38734	A set is succeeded by its successor. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)
Theorem	sucmapleftuniq 38735	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 38736*	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 38737	The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ pre 𝑁 ∈ V
Theorem	eupre2 38738*	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 38739*	Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))
Theorem	presucmap 38740	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 38741 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)
Theorem	preuniqval 38741*	Uniqueness/canonicity of pre. presucmap 38740 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 38742	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 38743	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 38744	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ⊆ 𝑁)
Theorem	preel 38745	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 38746*	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 38748 and the comment of dfec2 8648). 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 39210). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 38749) or to the range of a range Cartesian product of classes (cf. dfcoss4 38750), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 38748. Technically, we can define it via composition (dfcoss3 38749) or as the range of a range Cartesian product (dfcoss4 38750), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 39011, df-funALTV 39012) and disjoints (dfdisjs 39038, dfdisjs2 39039, df-disjALTV 39035, dfdisjALTV2 39044) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Definition	df-coels 38747	Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 38765. Possible definitions are the special cases of dfcoss3 38749 and dfcoss4 38750. (Contributed by Peter Mazsa, 20-Nov-2019.)
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
Theorem	dfcoss2 38748*	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 8648). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}
Theorem	dfcoss3 38749	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38746). (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
Theorem	dfcoss4 38750	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38746). (Contributed by Peter Mazsa, 12-Jul-2021.)
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)
Theorem	cosscnv 38751*	Class of cosets by the converse of 𝑅. (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
Theorem	coss1cnvres 38752*	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 38753*	Special case of coss1cnvres 38752. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(◡ E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))}
Theorem	cossex 38754	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 38755	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 38756	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 38757	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 38758	Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ Rel ≀ 𝑅
Theorem	relcoels 38759	Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ Rel ∼ 𝐴
Theorem	cossss 38760	Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Theorem	cosseq 38761	Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Theorem	cosseqi 38762	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 38763	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 38764*	The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Theorem	dfcoels 38765*	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 38766*	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 38767*	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 38768	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 38769	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 38770*	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 38771*	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 38772	The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ◡ ≀ 𝑅 = ≀ 𝑅
Theorem	br2coss 38773	Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Theorem	br1cossres 38774*	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 38775*	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 38776	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 38777*	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 38778*	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38777) is reflexive, see also refrelressn 38849. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Theorem	antisymressn 38779	Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38777) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)
⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Theorem	trressn 38780	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38777) is transitive, see also trrelressn 38912. (Contributed by Peter Mazsa, 16-Jun-2024.)
⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Theorem	relbrcoss 38781*	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 38782*	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 38783*	⟨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 38784*	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 38785*	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 38786*	⟨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 38787*	⟨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 38788	The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ dom ≀ 𝑅 = dom ◡𝑅
Theorem	dmcoss2 38789	The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ dom ≀ 𝑅 = ran 𝑅
Theorem	rncossdmcoss 38790	The range of cosets is the domain of them (this should be rncoss 5934 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
⊢ ran ≀ 𝑅 = dom ≀ 𝑅
Theorem	dm1cosscnvepres 38791	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 38792	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 38793*	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	eldmcoss2 38794	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴))
Theorem	eldm1cossres 38795*	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 38796*	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 38797	Lemma for the left side of the refrelcoss3 38798 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
Theorem	refrelcoss3 38798*	The class of cosets by 𝑅 is reflexive, see dfrefrel3 38841. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)
Theorem	refrelcoss2 38799	The class of cosets by 𝑅 is reflexive, see dfrefrel2 38840. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	symrelcoss3 38800	The class of cosets by 𝑅 is symmetric, see dfsymrel3 38879. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)