P. 376 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-epelg 37501	The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5543 and closed form of epeli 5542. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) TODO: move it to the main section after reordering to have brrelex1i 5696 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	bj-epelb 37502	Two classes are related by the membership relation if and only if they are related by the membership relation (i.e., the first is an element of the second) and the second is a set (hence so is the first). TODO: move to Main after reordering to have brrelex2i 5697 available. Check if it is shorter to prove bj-epelg 37501 first or bj-epelb 37502 first. (Contributed by BJ, 14-Jul-2023.)
⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
Theorem	bj-nsnid 37503	A set does not contain the singleton formed on it. More precisely, one can prove that a class contains the singleton formed on it if and only if it is proper and contains the empty set (since it is "the singleton formed on" any proper class, see snprc 4670): ⊢ ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴 → 𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴)
Theorem	bj-rdg0gALT 37504	Alternate proof of rdg0g 8386. More direct since it bypasses tz7.44-1 8365 and rdg0 8380 (and vtoclg 3516, vtoclga 3536). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
21.19.5.19  Axioms of separation and replacement This section proves basic relations among some standard axioms of set theory, in particular the axiom of separation (the universal closure of ax-sep 5240) and the version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function, bj-rep 37506. These axioms often appear (as specific instances) in the hypotheses of the theorems in this section.
Theorem	bj-axnul 37505*	Over the base theory ax-1 6-- ax-5 1924, the axiom of separation implies the weak emptyset axiom. By "weak emptyset axiom", we mean the axiom asserting existence of an empty set (which can be called "the" empty set when the axiom of extensionality ax-ext 2728 is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru 1989). This is the conclusion of bj-axnul 37505. Note that the weak emptyset axiom implies ⊢ (∃𝑥⊤ → ∃𝑦⊤) without DV conditions hence also the same statement as the weak emptyset axiom without DV conditions on 𝑥, but only on 𝑦, 𝑧. By "axiom of separation", we mean the universal closure of ax-sep 5240, simulated here by its instance with ⊥ substituted for 𝜑 (and with the variable used to assert existence in the weak emptyset axiom substituted for the containing set) as the hypothesis of bj-axnul 37505. In particular, the axiom of existence extru 1989 and the axiom of separation together imply the emptyset axiom (and conversely, the emptyset axiom implies the axiom of existence). Note: this theorem does not require a disjointness condition on 𝑦, 𝑧, although both axioms should be stated with all variables disjoint. This proof only uses an instance of the axiom of separation with a bounded formula, so is valid in a constructive setting (see the CZF section in the "Intuitionistic Logic Explorer" iset.mm). (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥))    ⇒   ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)
Theorem	bj-rep 37506*	Version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function from ax-rep 5221 (in the form of axrep6 5230). (Contributed by BJ, 14-Mar-2026.) The proof proves the statement without the DV condition on 𝑥, 𝜑, but the DV condition is added to this statement to show that this weaker version is sufficient. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
Theorem	bj-axseprep 37507*	Axiom of separation (universal closure of ax-sep 5240) from a weak form of the axiom of replacement requiring that the functional relation in it be a (total) function and the weak emptyset axiom (existence of an empty set provided existence of a set), as written in the theorem's hypotheses. This result shows that the weak emptyset axiom is not only the result of a cheap way to avoid an axiom redundancy (in this case, the existence axiom extru 1989) by adding it as an antecedent, but also permits to prove nontrivial results that hold in nonnecessarily nonempty universes. This proof is by cases so is not intuitionistic. The statement does not require a nonempty universe; most of the proof does not either, and the parts that do (e.g., near sb8ef 2380 and sbequ12r 2281 and eueq2 3667) could be reworked to avoid it. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1941. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)    &   ⊢ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓))    &   ⊢ (𝜓 ↔ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))    ⇒   ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))
Theorem	bj-axreprepsep 37508*	Strong axiom of replacement (universal closure of ax-rep 5221) from the axioms of separation and replacement as written in the theorem's hypotheses. The statement does not require a nonempty universe; most of the proof does not either, except for the use of 19.8a 2210, which could be removed by reworking the proof, since it is applied in a subexpression bound by the variable it introduces. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1941. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑠∀𝑦(𝑦 ∈ 𝑠 ↔ (𝑦 ∈ 𝑥 ∧ ∃𝑧𝜑))    &   ⊢ ∀𝑠(∀𝑦 ∈ 𝑠 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑠 𝜑))    ⇒   ⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))
21.19.5.20  Evaluation at a class This section treats the existing predicate Slot (df-slot 17194) as "evaluation at a class" and for the moment does not introduce new syntax for it.
Theorem	bj-evaleq 37509	Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17195 but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Theorem	bj-evalfun 37510	The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
⊢ Fun Slot 𝐴
Theorem	bj-evalfn 37511	The evaluation at a class is a function on the universal class. (General form of slotfn 17196). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
⊢ Slot 𝐴 Fn V
Theorem	bj-evalf 37512	The evaluation at a class is a function from the universal class into the universal class. (Contributed by BJ, 17-Mar-2026.)
⊢ Slot 𝐴:V⟶V
Theorem	bj-evalval 37513	Value of the evaluation at a class. Closed form of strfvnd 17197 and strfvn 17198. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)
⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))
Theorem	bj-evalid 37514	The evaluation at a set of the identity function is that set. General form of ndxarg 17208. The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
Theorem	bj-ndxarg 37515	Proof of ndxarg 17208 from bj-evalid 37514. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁
Theorem	bj-evalidval 37516	Closed general form of strndxid 17210. Both sides are equal to (𝐹‘𝐴) by bj-evalid 37514 and bj-evalval 37513 respectively, but bj-evalidval 37516 adds something to bj-evalid 37514 and bj-evalval 37513 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴‘𝐹))
21.19.5.21  Elementwise operations
Syntax	celwise 37517	Syntax for elementwise operations.
class elwise
Definition	df-elwise 37518*	Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 11093), so if 𝐴 and 𝐵 are sets of complex numbers, then (𝐴(elwise‘ + )𝐵) is the set of numbers of the form (𝑥 + 𝑦) with 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵. The set of odd natural numbers is (({2}(elwise‘ · )ℕ0)(elwise‘ + ){1}), or less formally 2ℕ0 + 1. (Contributed by BJ, 22-Dec-2021.)
⊢ elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)}))
21.19.5.22  Elementwise intersection (families of sets induced on a subset) Many kinds of structures are given by families of subsets of a given set: Moore collections (df-mre 17590), topologies (df-top 22927), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 34360), sigma rings, monotone classes, matroids/independent sets, bornologies, filters. There is a natural notion of structure induced on a subset. It is often given by an elementwise intersection, namely, the family of intersections of sets in the original family with the given subset. In this subsection, we define this notion and prove its main properties. Classical conditions on families of subsets include being nonempty, containing the whole set, containing the empty set, being stable under unions, intersections, subsets, supersets, (relative) complements. Therefore, we prove related properties for the elementwise intersection. We will call (𝑋 ↾t 𝐴) the elementwise intersection on the family 𝑋 by the class 𝐴. REMARK: many theorems are already in set.mm: "MM> SEARCH *rest* / JOIN".
Theorem	bj-rest00 37519	An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 17434. (Contributed by BJ, 27-Apr-2021.)
⊢ (∅ ↾t 𝐴) = ∅
Theorem	bj-restsn 37520	An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 37523 and bj-restsnid 37525. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})
Theorem	bj-restsnss 37521	Special case of bj-restsn 37520. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})
Theorem	bj-restsnss2 37522	Special case of bj-restsn 37520. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})
Theorem	bj-restsn0 37523	An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 37520 and bj-restsnss2 37522. TODO: this is restsn 23203. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅})
Theorem	bj-restsn10 37524	Special case of bj-restsn 37520, bj-restsnss 37521, and bj-rest10 37526. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → ({𝑋} ↾t ∅) = {∅})
Theorem	bj-restsnid 37525	The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 37520 and bj-restsnss 37521. (Contributed by BJ, 27-Apr-2021.)
⊢ ({𝐴} ↾t 𝐴) = {𝐴}
Theorem	bj-rest10 37526	An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 23202 and could replace it. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅}))
Theorem	bj-rest10b 37527	Alternate version of bj-rest10 37526. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅})
Theorem	bj-restn0 37528	An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ≠ ∅ → (𝑋 ↾t 𝐴) ≠ ∅))
Theorem	bj-restn0b 37529	Alternate version of bj-restn0 37528. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴 ∈ 𝑊) → (𝑋 ↾t 𝐴) ≠ ∅)
Theorem	bj-restpw 37530	The elementwise intersection on a powerset is the powerset of the intersection. This allows to prove for instance that the topology induced on a subset by the discrete topology is the discrete topology on that subset. See also restdis 23211 (which uses distop 23028 and restopn2 23210). (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))
Theorem	bj-rest0 37531	An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))
Theorem	bj-restb 37532	An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴)))
Theorem	bj-restv 37533	An elementwise intersection by a subset on a family containing the whole set contains the whole subset. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))
Theorem	bj-resta 37534	An elementwise intersection by a set on a family containing that set contains that set. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝐴 ∈ 𝑋 → 𝐴 ∈ (𝑋 ↾t 𝐴)))
Theorem	bj-restuni 37535	The union of an elementwise intersection by a set is equal to the intersection with that set of the union of the family. See also restuni 23195 and restuni2 23200. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))
Theorem	bj-restuni2 37536	The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23195 and restuni2 23200. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)
Theorem	bj-restreg 37537	A reformulation of the axiom of regularity using elementwise intersection. (RK: might have to be placed later since theorems in this section are to be moved early (in the section related to the algebra of sets).) (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∅ ∈ (𝐴 ↾t 𝐴))
21.19.5.23  Moore collections (complements)
Theorem	bj-raldifsn 37538*	All elements in a set satisfy a given property if and only if all but one satisfy that property and that one also does. Typically, this can be used for characterizations that are proved using different methods for a given element and for all others, for instance zero and nonzero numbers, or the empty set and nonempty sets. (Contributed by BJ, 7-Dec-2021.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ∧ 𝜓)))
Theorem	bj-0int 37539*	If 𝐴 is a collection of subsets of 𝑋, like a Moore collection or a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵: the LHS singles out the empty intersection (the empty intersection relative to 𝑋 is 𝑋 and the intersection of a nonempty family of subsets of 𝑋 is included in 𝑋, so there is no need to intersect it with 𝑋). In typical applications, 𝐵 is 𝐴 itself. (Contributed by BJ, 7-Dec-2021.)
⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
Theorem	bj-mooreset 37540*	A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 37542 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 37543. Note that the closed sets of a topology form a Moore collection, so a topology is a set, and this remark also applies to many other families of sets (namely, as soon as the whole set is required to be a set of the family, then the associated kind of family has no proper classes: that this condition suffices to impose sethood can be seen in this proof, which relies crucially on uniexr 7735). Note: if, in the above predicate, we substitute 𝒫 𝑋 for 𝐴, then the last ∈ 𝒫 𝑋 could be weakened to ⊆ 𝑋, and then the predicate would be obviously satisfied since ⊢ ∪ 𝒫 𝑋 = 𝑋 (unipw 5411), making 𝒫 𝑋 a Moore collection in this weaker sense, for any class 𝑋, even proper, but the addition of this single case does not add anything interesting. Instead, we have the biconditional bj-discrmoore 37549. (Contributed by BJ, 8-Dec-2021.)
⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
Syntax	cmoore 37541	Syntax for the class of Moore collections.
class Moore
Definition	df-bj-moore 37542*	Define the class of Moore collections. This is indeed the class of all Moore collections since these all are sets, as proved in bj-mooreset 37540, and as illustrated by the lack of sethood condition in bj-ismoore 37543. This is to df-mre 17590 (defining Moore) what df-top 22927 (defining Top) is to df-topon 22944 (defining TopOn). For the sake of consistency, the function defined at df-mre 17590 should be denoted by "MooreOn". Note: df-mre 17590 singles out the empty intersection. This is not necessary. It could be written instead ⊢ Moore = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝒫 𝑥 ∣ ∀𝑧 ∈ 𝒫 𝑦(𝑥 ∩ ∩ 𝑧) ∈ 𝑦}) and the equivalence of both definitions is proved by bj-0int 37539. There is no added generality in defining a "Moore predicate" for arbitrary classes, since a Moore class satisfying such a predicate is automatically a set (see bj-mooreset 37540). TODO: move to the main section. For many families of sets, one can define both the function associating to each set the set of families of that kind on it (like df-mre 17590 and df-topon 22944) or the class of all families of that kind, independent of a base set (like df-bj-moore 37542 or df-top 22927). In general, the former will be more useful and the extra generality of the latter is not necessary. Moore collections, however, are particular in that they are more ubiquitous and are used in a wide variety of applications (for many families of sets, the family of families of a given kind is often a Moore collection, for instance). Therefore, in the case of Moore families, having both definitions is useful. (Contributed by BJ, 27-Apr-2021.)
⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
Theorem	bj-ismoore 37543*	Characterization of Moore collections. Note that there is no sethood hypothesis on 𝐴: it is implied by either side (this is obvious for the LHS, and is the content of bj-mooreset 37540 for the RHS). (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
Theorem	bj-ismoored0 37544	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)
Theorem	bj-ismoored 37545	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ Moore)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
Theorem	bj-ismoored2 37546	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ Moore)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    ⇒   ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)
Theorem	bj-ismooredr 37547*	Sufficient condition to be a Moore collection. Note that there is no sethood hypothesis on 𝐴: it is a consequence of the only hypothesis. (Contributed by BJ, 9-Dec-2021.)
⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ Moore)
Theorem	bj-ismooredr2 37548*	Sufficient condition to be a Moore collection (variant of bj-ismooredr 37547 singling out the empty intersection). Note that there is no sethood hypothesis on 𝐴: it is a consequence of the first hypothesis. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ Moore)
Theorem	bj-discrmoore 37549	The powerclass 𝒫 𝐴 is a Moore collection if and only if 𝐴 is a set. It is then called the discrete Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)
Theorem	bj-0nmoore 37550	The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ ¬ ∅ ∈ Moore
Theorem	bj-snmoore 37551	A singleton is a Moore collection. See bj-snmooreb 37552 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)
Theorem	bj-snmooreb 37552	A singleton is a Moore collection, biconditional version. (Contributed by BJ, 9-Dec-2021.) (Proof shortened by BJ, 10-Apr-2024.)
⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)
Theorem	bj-prmoore 37553	A pair formed of two nested sets is a Moore collection. (Note that in the statement, if 𝐵 is a proper class, we are in the case of bj-snmoore 37551). A direct consequence is ⊢ {∅, 𝐴} ∈ Moore. More generally, any nonempty well-ordered chain of sets that is a set is a Moore collection. We also have the biconditional ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → ({𝐴, 𝐵} ∈ Moore ↔ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))). (Contributed by BJ, 11-Apr-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → {𝐴, 𝐵} ∈ Moore)
21.19.5.24  Maps-to notation for functions with three arguments
Theorem	bj-0nelmpt 37554	The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	bj-mptval 37555	Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))
Theorem	bj-dfmpoa 37556*	An equivalent definition of df-mpo 7390. (Contributed by BJ, 30-Dec-2020.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Theorem	bj-mpomptALT 37557*	Alternate proof of mpompt 7499. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)    ⇒   ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Syntax	cmpt3 37558	Syntax for maps-to notation for functions with three arguments.
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
Definition	df-bj-mpt3 37559*	Define maps-to notation for functions with three arguments. See df-mpt 5176 and df-mpo 7390 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpoa 37556. (Contributed by BJ, 11-Apr-2020.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
21.19.5.25  Currying Currying and uncurrying. See also df-cur 8235 and df-unc 8236. Contrary to these, the definitions in this section are parameterized.
Syntax	csethom 37560	Syntax for the set of set morphisms.
class Set⟶
Definition	df-bj-sethom 37561*	Define the set of functions (morphisms of sets) between two sets. Same as df-map 8798 with arguments swapped. TODO: prove the same staple lemmas as for ↑m. Remark: one may define Set⟶ = (𝑥 ∈ dom Struct , 𝑦 ∈ dom Struct ↦ {𝑓 ∣ 𝑓:(Base‘𝑥)⟶(Base‘𝑦)}) so that for morphisms between other structures, one could write ... = {𝑓 ∈ (𝑥 Set⟶ 𝑦) ∣ ...}. (Contributed by BJ, 11-Apr-2020.)
⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
Syntax	ctophom 37562	Syntax for the set of topological morphisms.
class Top⟶
Definition	df-bj-tophom 37563*	Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 23260 (which is in terms of topologies instead of topological spaces). (Contributed by BJ, 10-Feb-2022.)
⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
Syntax	cmgmhom 37564	Syntax for the set of magma morphisms.
class Mgm⟶
Definition	df-bj-mgmhom 37565*	Define the set of magma morphisms between two magmas. If domain and codomain are semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. (Contributed by BJ, 10-Feb-2022.)
⊢ Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))})
Syntax	ctopmgmhom 37566	Syntax for the set of topological magma morphisms.
class TopMgm⟶
Definition	df-bj-topmgmhom 37567*	Define the set of topological magma morphisms (continuous magma morphisms) between two topological magmas. If domain and codomain are topological semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. This definition is currently stated with topological monoid domain and codomain, since topological magmas are currently not defined in set.mm. (Contributed by BJ, 10-Feb-2022.)
⊢ TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
Syntax	ccur- 37568	Syntax for the parameterized currying function.
class curry_
Definition	df-bj-cur 37569*	Define currying. See also df-cur 8235. (Contributed by BJ, 11-Apr-2020.)
⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
Syntax	cunc- 37570	Notation for the parameterized uncurrying function.
class uncurry_
Definition	df-bj-unc 37571*	Define uncurrying. See also df-unc 8236. (Contributed by BJ, 11-Apr-2020.)
⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
21.19.5.26  Setting components of extensible structures Groundwork for changing the definition, syntax and token for component-setting in extensible structures. See https://github.com/metamath/set.mm/issues/2401
Syntax	cstrset 37572	Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆
Definition	df-strset 37573	Component-setting in extensible structures. Define the extensible structure [𝐵 / 𝐴]struct𝑆, which is like the extensible structure 𝑆 except that the value 𝐵 has been put in the slot 𝐴 (replacing the current value if there was already one). In such expressions, 𝐴 is generally substituted for slot mnemonics like Base or +g or dist. The V in this definition was chosen to be closer to df-sets 17176, but since extensible structures are functions on ℕ, it will be more natural to replace it with ℕ when df-strset 37573 becomes the main definition. (Contributed by BJ, 13-Feb-2022.)
⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
Theorem	setsstrset 37574	Relation between df-sets 17176 and df-strset 37573. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(𝐴‘ndx), 𝐵⟩))
21.19.6  Extended real and complex numbers, real and complex projective lines In this section, we indroduce several supersets of the set ℝ of real numbers and the set ℂ of complex numbers. Once they are given their usual topologies, which are locally compact, both topological spaces have a one-point compactification. They are denoted by ℝ̂ and ℂ̂ respectively, defined in df-bj-cchat 37673 and df-bj-rrhat 37675, and the point at infinity is denoted by ∞, defined in df-bj-infty 37671. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 37669 (already defined as ℝ*, see df-xr 11210) and ℂ̅, defined in df-bj-ccbar 37656. Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to ℂ a "point at infinity at the end of each ray with origin at 0". Although ℂ̅ is not an important object in itself, the motivation for introducing it is to provide a common superset to both ℝ̅ and ℂ and to define algebraic operations (addition, opposite, multiplication, inverse) as widely as reasonably possible. Mathematically, ℂ̅ is the quotient of ((ℂ × ℝ≥0) ∖ {⟨0, 0⟩}) by the diagonal multiplicative action of ℝ>0 (think of the closed "northern hemisphere" in ℝ^3 identified with (ℂ × ℝ), that each open ray from 0 included in the closed northern half-space intersects exactly once). Since in set.mm, we want to have a genuine inclusion ℂ ⊆ ℂ̅, we instead define ℂ̅ as the (disjoint) union of ℂ with a circle at infinity denoted by ℂ∞. To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in ℂ∞. Thanks to this framework, one has the genuine inclusions ℝ ⊆ ℝ̅ and ℝ ⊆ ℝ̂ and similarly ℂ ⊆ ℂ̅ and ℂ ⊆ ℂ̂. Furthermore, one has ℝ ⊆ ℂ as well as ℝ̅ ⊆ ℂ̅ and ℝ̂ ⊆ ℂ̂. Furthermore, we define the main algebraic operations on (ℂ̅ ∪ ℂ̂), which is not very mathematical, but "overloads" the operations, so that one can use the same notation in all cases.
21.19.6.1  Complements on class abstractions of ordered pairs and binary relations
Theorem	bj-nfald 37575	Variant of nfald 2354. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	bj-nfexd 37576	Variant of nfexd 2355. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	cgsex2gd 37577*	Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) Adapt cgsex2g 3493 to deduction form. (Revised by BJ, 28-Mar-2026.) Do not use cgsex2g 3493. (Proof modification is discouraged.)
⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) ↔ 𝜃))
Theorem	copsex2gd 37578*	Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5458 to reduce axiom usage. (Revised by SN, 1-Sep-2024.) Adapt copsex2g 5456 $p to deduction form. (Revised by BJ, 28-Mar-2026.) Do not use copsex2g 5456. (Proof modification is discouraged.)
⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Theorem	copsex2d 37579*	Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Theorem	copsex2b 37580*	Biconditional form of copsex2d 37579. TODO: prove a relative version, that is, with ∃𝑥 ∈ 𝑉∃𝑦 ∈ 𝑊...(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊). (Contributed by BJ, 27-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Theorem	opelopabd 37581*	Membership of an ordered pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
Theorem	opelopabb 37582*	Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Theorem	opelopabbv 37583*	Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Theorem	bj-opelrelex 37584	The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5692, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-opelresdm 37585	If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5964. (Contributed by BJ, 25-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
Theorem	bj-brresdm 37586	If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5965 and brrelex1 5693. Remark: there are many pairs like bj-opelresdm 37585 / bj-brresdm 37586, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37585 / brrelex12 5692 or the opelopabg 5502 / brabg 5503 family). They are straightforwardly equivalent by df-br 5095. The latter is indeed a very direct definition, introducing a "shorthand", and barely necessary, were it not for the frequency of the expression 𝐴𝑅𝐵. Therefore, in the spirit of "definitions are here to be used", most theorems, apart from the most elementary ones, should only have the "br" version, not the "opel" one. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)
Theorem	brabd0 37587*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))
Theorem	brabd 37588*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))
Theorem	bj-brab2a1 37589*	"Unbounded" version of brab2a 5733. (Contributed by BJ, 25-Dec-2023.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))
21.19.6.2  Identity relation (complements) Complements on the identity relation.
Theorem	bj-opabssvv 37590*	A variant of relopabiv 5786 (which could be proved from it, similarly to relxp 5658 from xpss 5656). (Contributed by BJ, 28-Dec-2023.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
Theorem	bj-funidres 37591	The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.) TODO: relabel funi 6542 to funid.
⊢ Fun ( I ↾ 𝑉)
Theorem	bj-opelidb 37592	Characterization of the ordered pair elements of the identity relation. Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than ⊤ which already appears in the proof. Here for instance this could be the definition I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
Theorem	bj-opelidb1 37593	Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 37592 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-inexeqex 37594	Lemma for bj-opelid 37596 (but not specific to the identity relation): if the intersection of two classes is a set and the two classes are equal, then both are sets (all three classes are equal, so they all belong to 𝑉, but it is more convenient to have V in the consequent for theorems using it). (Contributed by BJ, 27-Dec-2023.)
⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-elsn0 37595	If the intersection of two classes is a set, then these classes are equal if and only if one is an element of the singleton formed on the other. Stronger form of elsng 4590 and elsn2g 4617 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	bj-opelid 37596	Characterization of the ordered pair elements of the identity relation when the intersection of their components are sets. Note that the antecedent is more general than either component being a set. (Contributed by BJ, 29-Mar-2020.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
Theorem	bj-ideqg 37597	Characterization of the classes related by the identity relation when their intersection is a set. Note that the antecedent is more general than either class being a set. (Contributed by NM, 30-Apr-2004.) Weaken the antecedent to sethood of the intersection. (Revised by BJ, 24-Dec-2023.) TODO: replace ideqg 5816, or at least prove ideqg 5816 from it.
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqgALT 37598	Alternate proof of bj-ideqg 37597 from brabga 5498 instead of bj-opelid 37596 itself proved from bj-opelidb 37592. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqb 37599	Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-idres 37600	Alternate expression for the restricted identity relation. The advantage of that expression is to expose it as a "bounded" class, being included in the Cartesian square of the restricting class. (Contributed by BJ, 27-Dec-2023.) This is an alternate of idinxpresid 6027 (see idinxpres 6026). See also elrid 6025 and elidinxp 6023. (Proof modification is discouraged.)
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))