P. 372 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37101-37200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-epelg 37101	The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5519 and closed form of epeli 5518. (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 5672 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	bj-epelb 37102	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 5673 available. Check if it is shorter to prove bj-epelg 37101 first or bj-epelb 37102 first. (Contributed by BJ, 14-Jul-2023.)
⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
Theorem	bj-nsnid 37103	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 37104	Alternate proof of rdg0g 8346. More direct since it bypasses tz7.44-1 8325 and rdg0 8340 (and vtoclg 3509, vtoclga 3532). (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  Evaluation at a class This section treats the existing predicate Slot (df-slot 17090) as "evaluation at a class" and for the moment does not introduce new syntax for it.
Theorem	bj-evaleq 37105	Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17091 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 37106	The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
⊢ Fun Slot 𝐴
Theorem	bj-evalfn 37107	The evaluation at a class is a function on the universal class. (General form of slotfn 17092). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
⊢ Slot 𝐴 Fn V
Theorem	bj-evalval 37108	Value of the evaluation at a class. (Closed form of strfvnd 17093 and strfvn 17094). (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 37109	The evaluation at a set of the identity function is that set. (General form of ndxarg 17104.) 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 37110	Proof of ndxarg 17104 from bj-evalid 37109. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁
Theorem	bj-evalidval 37111	Closed general form of strndxid 17106. Both sides are equal to (𝐹‘𝐴) by bj-evalid 37109 and bj-evalval 37108 respectively, but bj-evalidval 37111 adds something to bj-evalid 37109 and bj-evalval 37108 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))
21.19.5.20  Elementwise operations
Syntax	celwise 37112	Syntax for elementwise operations.
class elwise
Definition	df-elwise 37113*	Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 11033), 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)}))
21.19.5.21  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 17485), topologies (df-top 22807), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 34117), 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 37114	An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 17330. (Contributed by BJ, 27-Apr-2021.)
⊢ (∅ ↾t 𝐴) = ∅
Theorem	bj-restsn 37115	An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 37118 and bj-restsnid 37120. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})
Theorem	bj-restsnss 37116	Special case of bj-restsn 37115. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})
Theorem	bj-restsnss2 37117	Special case of bj-restsn 37115. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})
Theorem	bj-restsn0 37118	An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 37115 and bj-restsnss2 37117. TODO: this is restsn 23083. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅})
Theorem	bj-restsn10 37119	Special case of bj-restsn 37115, bj-restsnss 37116, and bj-rest10 37121. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → ({𝑋} ↾t ∅) = {∅})
Theorem	bj-restsnid 37120	The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 37115 and bj-restsnss 37116. (Contributed by BJ, 27-Apr-2021.)
⊢ ({𝐴} ↾t 𝐴) = {𝐴}
Theorem	bj-rest10 37121	An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 23082 and could replace it. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅}))
Theorem	bj-rest10b 37122	Alternate version of bj-rest10 37121. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅})
Theorem	bj-restn0 37123	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 37124	Alternate version of bj-restn0 37123. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴 ∈ 𝑊) → (𝑋 ↾t 𝐴) ≠ ∅)
Theorem	bj-restpw 37125	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 23091 (which uses distop 22908 and restopn2 23090). (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))
Theorem	bj-rest0 37126	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 37127	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 37128	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 37129	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 37130	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 23075 and restuni2 23080. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))
Theorem	bj-restuni2 37131	The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23075 and restuni2 23080. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)
Theorem	bj-restreg 37132	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))
21.19.5.22  Moore collections (complements)
Theorem	bj-raldifsn 37133*	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 37134*	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 37135*	A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 37137 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 37138. 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 7696). 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 5391), 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 37144. (Contributed by BJ, 8-Dec-2021.)
⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
Syntax	cmoore 37136	Syntax for the class of Moore collections.
class Moore
Definition	df-bj-moore 37137*	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 37135, and as illustrated by the lack of sethood condition in bj-ismoore 37138. This is to df-mre 17485 (defining Moore) what df-top 22807 (defining Top) is to df-topon 22824 (defining TopOn). For the sake of consistency, the function defined at df-mre 17485 should be denoted by "MooreOn". Note: df-mre 17485 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 37134. 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 37135). 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 17485 and df-topon 22824) or the class of all families of that kind, independent of a base set (like df-bj-moore 37137 or df-top 22807). 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 37138*	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 37135 for the RHS). (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
Theorem	bj-ismoored0 37139	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)
Theorem	bj-ismoored 37140	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 37141	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 37142*	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 37143*	Sufficient condition to be a Moore collection (variant of bj-ismooredr 37142 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 37144	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 37145	The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ ¬ ∅ ∈ Moore
Theorem	bj-snmoore 37146	A singleton is a Moore collection. See bj-snmooreb 37147 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)
Theorem	bj-snmooreb 37147	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 37148	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 37146). 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)
21.19.5.23  Maps-to notation for functions with three arguments
Theorem	bj-0nelmpt 37149	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 37150	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 37151*	An equivalent definition of df-mpo 7351. (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 37152*	Alternate proof of mpompt 7460. (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 37153	Syntax for maps-to notation for functions with three arguments.
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
Definition	df-bj-mpt3 37154*	Define maps-to notation for functions with three arguments. See df-mpt 5173 and df-mpo 7351 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpoa 37151. (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)}
21.19.5.24  Currying Currying and uncurrying. See also df-cur 8197 and df-unc 8198. Contrary to these, the definitions in this section are parameterized.
Syntax	csethom 37155	Syntax for the set of set morphisms.
class Set⟶
Definition	df-bj-sethom 37156*	Define the set of functions (morphisms of sets) between two sets. Same as df-map 8752 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 37157	Syntax for the set of topological morphisms.
class Top⟶
Definition	df-bj-tophom 37158*	Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 23140 (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 37159	Syntax for the set of magma morphisms.
class Mgm⟶
Definition	df-bj-mgmhom 37160*	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 37161	Syntax for the set of topological magma morphisms.
class TopMgm⟶
Definition	df-bj-topmgmhom 37162*	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- 37163	Syntax for the parameterized currying function.
class curry_
Definition	df-bj-cur 37164*	Define currying. See also df-cur 8197. (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- 37165	Notation for the parameterized uncurrying function.
class uncurry_
Definition	df-bj-unc 37166*	Define uncurrying. See also df-unc 8198. (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))))
21.19.5.25  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 37167	Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆
Definition	df-strset 37168	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 17072, but since extensible structures are functions on ℕ, it will be more natural to replace it with ℕ when df-strset 37168 becomes the main definition. (Contributed by BJ, 13-Feb-2022.)
⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
Theorem	setsstrset 37169	Relation between df-sets 17072 and df-strset 37168. 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 37266 and df-bj-rrhat 37268, and the point at infinity is denoted by ∞, defined in df-bj-infty 37264. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 37262 (already defined as ℝ*, see df-xr 11147) and ℂ̅, defined in df-bj-ccbar 37249. 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 37170	Variant of nfald 2329. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	bj-nfexd 37171	Variant of nfexd 2330. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	copsex2d 37172*	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 37173*	Biconditional form of copsex2d 37172. 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 37174*	Membership of an ordere 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 37175*	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 37176*	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 37177	The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5668, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-opelresdm 37178	If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5934. (Contributed by BJ, 25-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
Theorem	bj-brresdm 37179	If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5935 and brrelex1 5669. Remark: there are many pairs like bj-opelresdm 37178 / bj-brresdm 37179, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37178 / brrelex12 5668 or the opelopabg 5478 / brabg 5479 family). They are straightforwardly equivalent by df-br 5092. 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 37180*	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 37181*	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 37182*	"Unbounded" version of brab2a 5709. (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 37183*	A variant of relopabiv 5760 (which could be proved from it, similarly to relxp 5634 from xpss 5632). (Contributed by BJ, 28-Dec-2023.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
Theorem	bj-funidres 37184	The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.) TODO: relabel funi 6513 to funid.
⊢ Fun ( I ↾ 𝑉)
Theorem	bj-opelidb 37185	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 37186	Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 37185 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 37187	Lemma for bj-opelid 37189 (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 37188	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 37189	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 37190	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 5791, or at least prove ideqg 5791 from it.
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqgALT 37191	Alternate proof of bj-ideqg 37190 from brabga 5474 instead of bj-opelid 37189 itself proved from bj-opelidb 37185. (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 37192	Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-idres 37193	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 5997 (see idinxpres 5996). See also elrid 5995 and elidinxp 5993. (Proof modification is discouraged.)
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Theorem	bj-opelidres 37194	Characterization of the ordered pairs in the restricted identity relation when the intersection of their component belongs to the restricting class. TODO: prove bj-idreseq 37195 from it. (Contributed by BJ, 29-Mar-2020.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
Theorem	bj-idreseq 37195	Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37190 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37195. This is a strengthening of resieq 5939 which should be proved from it (note that currently, resieq 5939 relies on ideq 5792). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ⊢ ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → ...). (Contributed by BJ, 25-Dec-2023.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-idreseqb 37196	Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))
Theorem	bj-ideqg1 37197	For sets, the identity relation is the same thing as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ, 24-Dec-2023.) TODO: delete once bj-ideqg 37190 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqg1ALT 37198	Alternate proof of bj-ideqg1 using brabga 5474 instead of the "unbounded" version bj-brab2a1 37182 or brab2a 5709. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) TODO: delete once bj-ideqg 37190 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-opelidb1ALT 37199	Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-elid3 37200	Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.)
⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)