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Theorem List for Metamath Proof Explorer - 34801-34900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-epelb 34801 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 5584 available. Check if it is shorter to prove bj-epelg 34800 first or bj-epelb 34801 first. (Contributed by BJ, 14-Jul-2023.)
(𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))

Theorembj-nsnid 34802 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 4614): ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
(𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)

20.15.5.17  Evaluation

This section treats the existing predicate Slot (df-slot 16560) as "evaluation at a class" and for the moment does not introduce new syntax for it.

Theorembj-evaleq 34803 Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16561 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)

Theorembj-evalfun 34804 The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
Fun Slot 𝐴

Theorembj-evalfn 34805 The evaluation at a class is a function on the universal class. (General form of slotfn 16574). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
Slot 𝐴 Fn V

Theorembj-evalval 34806 Value of the evaluation at a class. (Closed form of strfvnd 16575 and strfvn 16578). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)
(𝐹𝑉 → (Slot 𝐴𝐹) = (𝐹𝐴))

Theorembj-evalid 34807 The evaluation at a set of the identity function is that set. (General form of ndxarg 16581.) 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.)
((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Theorembj-ndxarg 34808 Proof of ndxarg 16581 from bj-evalid 34807. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theorembj-evalidval 34809 Closed general form of strndxid 16583. Both sides are equal to (𝐹𝐴) by bj-evalid 34807 and bj-evalval 34806 respectively, but bj-evalidval 34809 adds something to bj-evalid 34807 and bj-evalval 34806 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))

20.15.5.18  Elementwise operations

Syntaxcelwise 34810 Syntax for elementwise operations.
class elwise

Definitiondf-elwise 34811* Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 10619), 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 𝑥𝐴 and 𝑦𝐵. 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 ∣ ∃𝑢𝑥𝑣𝑦 𝑧 = (𝑢𝑜𝑣)}))

20.15.5.19  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 16930), topologies (df-top 21609), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 31610), 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 (𝑋t 𝐴) the elementwise intersection on the family 𝑋 by the class 𝐴.

REMARK: many theorems are already in set.mm: "MM> SEARCH *rest* / JOIN".

Theorembj-rest00 34812 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 16776. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅

Theorembj-restsn 34813 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 34816 and bj-restsnid 34818. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Theorembj-restsnss 34814 Special case of bj-restsn 34813. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})

Theorembj-restsnss2 34815 Special case of bj-restsn 34813. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})

Theorembj-restsn0 34816 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 34813 and bj-restsnss2 34815. TODO: this is restsn 21885. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theorembj-restsn10 34817 Special case of bj-restsn 34813, bj-restsnss 34814, and bj-rest10 34819. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})

Theorembj-restsnid 34818 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 34813 and bj-restsnss 34814. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}

Theorembj-rest10 34819 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 21884 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))

Theorembj-rest10b 34820 Alternate version of bj-rest10 34819. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})

Theorembj-restn0 34821 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Theorembj-restn0b 34822 Alternate version of bj-restn0 34821. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)

Theorembj-restpw 34823 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 21893 (which uses distop 21710 and restopn2 21892). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Theorembj-rest0 34824 An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))

Theorembj-restb 34825 An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))

Theorembj-restv 34826 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 𝑋𝑋) → 𝐴 ∈ (𝑋t 𝐴))

Theorembj-resta 34827 An elementwise intersection by a set on a family containing that set contains that set. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝐴𝑋𝐴 ∈ (𝑋t 𝐴)))

Theorembj-restuni 34828 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 21877 and restuni2 21882. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Theorembj-restuni2 34829 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 21877 and restuni2 21882. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Theorembj-restreg 34830 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.)
((𝐴𝑉𝐴 ≠ ∅) → ∅ ∈ (𝐴t 𝐴))

20.15.5.20  Moore collections (complements)

Theorembj-raldifsn 34831* 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 → (𝜑𝜓))       (𝐵𝐴 → (∀𝑥𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑𝜓)))

Theorembj-0int 34832* 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 → ((𝑋𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅}) 𝑥𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 𝑥) ∈ 𝐵))

Theorembj-mooreset 34833* A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 34835 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 34836.

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 7491).

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 5316) , 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 34842. (Contributed by BJ, 8-Dec-2021.)

(∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)

Syntaxcmoore 34834 Syntax for the class of Moore collections.
class Moore

Definitiondf-bj-moore 34835* 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 34833, and as illustrated by the lack of sethood condition in bj-ismoore 34836.

This is to df-mre 16930 (defining Moore) what df-top 21609 (defining Top) is to df-topon 21626 (defining TopOn).

For the sake of consistency, the function defined at df-mre 16930 should be denoted by "MooreOn".

Note: df-mre 16930 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 34832.

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 34833).

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 16930 and df-topon 21626) or the class of all families of that kind, independent of a base set (like df-bj-moore 34835 or df-top 21609). 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}

Theorembj-ismoore 34836* 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 34833 for the RHS). (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)

Theorembj-ismoored0 34837 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore 𝐴𝐴)

Theorembj-ismoored 34838 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)       (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)

Theorembj-ismoored2 34839 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ≠ ∅)       (𝜑 𝐵𝐴)

Theorembj-ismooredr 34840* 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.)
((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)       (𝜑𝐴Moore)

Theorembj-ismooredr2 34841* Sufficient condition to be a Moore collection (variant of bj-ismooredr 34840 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 𝐴𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑥 ≠ ∅)) → 𝑥𝐴)       (𝜑𝐴Moore)

Theorembj-discrmoore 34842 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 𝐴Moore)

Theorembj-0nmoore 34843 The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
¬ ∅ ∈ Moore

Theorembj-snmoore 34844 A singleton is a Moore collection. See bj-snmooreb 34845 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
(𝐴𝑉 → {𝐴} ∈ Moore)

Theorembj-snmooreb 34845 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)

Theorembj-prmoore 34846 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 34844). 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 ((𝐴𝐵) ∈ 𝑉 ({𝐴, 𝐵} ∈ Moore ↔ (𝐴𝐵𝐵𝐴))). (Contributed by BJ, 11-Apr-2024.)

((𝐴𝑉𝐴𝐵) → {𝐴, 𝐵} ∈ Moore)

20.15.5.21  Maps-to notation for functions with three arguments

Theorembj-0nelmpt 34847 The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
¬ ∅ ∈ (𝑥𝐴𝐵)

Theorembj-mptval 34848 Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)
𝑥𝐴       (∀𝑥𝐴 𝐵𝑉 → (𝑋𝐴 → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌)))

Theorembj-dfmpoa 34849* An equivalent definition of df-mpo 7162. (Contributed by BJ, 30-Dec-2020.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Theorembj-mpomptALT 34850* Alternate proof of mpompt 7267. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Syntaxcmpt3 34851 Syntax for maps-to notation for functions with three arguments.
class (𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷)

Definitiondf-bj-mpt3 34852* Define maps-to notation for functions with three arguments. See df-mpt 5118 and df-mpo 7162 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpoa 34849. (Contributed by BJ, 11-Apr-2020.)
(𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}

20.15.5.22  Currying

Currying and uncurrying. See also df-cur 7950 and df-unc 7951. Contrary to these, the definitions in this section are parameterized.

Syntaxcsethom 34853 Syntax for the set of set morphisms.
class Set

Definitiondf-bj-sethom 34854* Define the set of functions (morphisms of sets) between two sets. Same as df-map 8425 with arguments swapped. TODO: prove the same staple lemmas as for m.

Remark: one may define Set⟶ = (𝑥 ∈ dom Struct , 𝑦 ∈ dom Struct ↦ {𝑓𝑓:(Base‘𝑥)⟶(Base‘𝑦)}) so that for morphisms between other structures, one could write ... = {𝑓 ∈ (𝑥 Set𝑦) ∣ ...}.

(Contributed by BJ, 11-Apr-2020.)

Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑥𝑦})

Syntaxctophom 34855 Syntax for the set of topological morphisms.
class Top

Definitiondf-bj-tophom 34856* Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 21942 (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)(𝑓𝑢) ∈ (TopOpen‘𝑥)})

Syntaxcmgmhom 34857 Syntax for the set of magma morphisms.
class Mgm

Definitiondf-bj-mgmhom 34858* 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(+g𝑥)𝑣)) = ((𝑓𝑢)(+g𝑦)(𝑓𝑣))})

Syntaxctopmgmhom 34859 Syntax for the set of topological magma morphisms.
class TopMgm

Definitiondf-bj-topmgmhom 34860* 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 Top𝑦) ∩ (𝑥 Mgm𝑦)))

Syntaxccur- 34861 Syntax for the parameterized currying function.
class curry_

Definitiondf-bj-cur 34862* Define currying. See also df-cur 7950. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Syntaxcunc- 34863 Notation for the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-unc 34864* Define uncurrying. See also df-unc 7951. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

20.15.5.23  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

Syntaxcstrset 34865 Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆

Definitiondf-strset 34866 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 16563, but since extensible structures are functions on , it will be more natural to replace it with when df-strset 34866 becomes the main definition. (Contributed by BJ, 13-Feb-2022.)
[𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})

Theoremsetsstrset 34867 Relation between df-sets 16563 and df-strset 34866. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.)
((𝑆𝑉𝐵𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(𝐴‘ndx), 𝐵⟩))

20.15.6  Extended real and complex numbers, real and complex projective lines

In this section, we indroduce several supersets of the set of real numbers and the set 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 34964 and df-bj-rrhat 34966, and the point at infinity is denoted by , defined in df-bj-infty 34962.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 34960 (already defined as *, see df-xr 10731) and ℂ̅, defined in df-bj-ccbar 34947.

Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to 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 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 >0 (think of the closed "northern hemisphere" in ^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 with a circle at infinity denoted by . To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in .

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.

20.15.6.1  Complements on class abstractions of ordered pairs and binary relations

Theorembj-nfald 34868 Variant of nfald 2337. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theorembj-nfexd 34869 Variant of nfexd 2338. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremcopsex2d 34870* Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))

Theoremcopsex2b 34871* Biconditional form of copsex2d 34870. TODO: prove a relative version, that is, with 𝑥𝑉𝑦𝑊...(𝐴𝑉𝐵𝑊). (Contributed by BJ, 27-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))

Theoremopelopabd 34872* 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)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))

Theoremopelopabb 34873* 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 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))

Theoremopelopabbv 34874* Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
(𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))

Theorembj-opelrelex 34875 The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5579, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-opelresdm 34876 If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5835. (Contributed by BJ, 25-Dec-2023.)
(⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)

Theorembj-brresdm 34877 If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5836 and brrelex1 5580.

Remark: there are many pairs like bj-opelresdm 34876 / bj-brresdm 34877, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34876 / brrelex12 5579 or the opelopabg 5400 / brabg 5401 family). They are straightforwardly equivalent by df-br 5038. 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(𝑅𝑋)𝐵𝐴𝑋)

Theorembrabd0 34878* 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)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (𝐴𝑅𝐵𝜒))

Theorembrabd 34879* 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.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (𝐴𝑅𝐵𝜒))

Theorembj-brab2a1 34880* "Unbounded" version of brab2a 5619. (Contributed by BJ, 25-Dec-2023.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))

20.15.6.2  Identity relation (complements)

Complements on the identity relation.

Theorembj-opabssvv 34881* A variant of relopabiv 5668 (which could be proved from it, similarly to relxp 5547 from xpss 5545). (Contributed by BJ, 28-Dec-2023.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)

Theorembj-funidres 34882 The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.)

TODO: relabel funi 6373 to funid.

Fun ( I ↾ 𝑉)

Theorembj-opelidb 34883 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))

Theorembj-opelidb1 34884 Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 34883 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Theorembj-inexeqex 34885 Lemma for bj-opelid 34887 (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.)
(((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-elsn0 34886 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 4540 and elsn2g 4564 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Theorembj-opelid 34887 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.)
((𝐴𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))

Theorembj-ideqg 34888 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 5698, or at least prove ideqg 5698 from it.

((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Theorembj-ideqgALT 34889 Alternate proof of bj-ideqg 34888 from brabga 5396 instead of bj-opelid 34887 itself proved from bj-opelidb 34883. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Theorembj-ideqb 34890 Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
(𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Theorembj-idres 34891 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 5893 (see idinxpres 5892). See also elrid 5891 and elidinxp 5889. (Proof modification is discouraged.)

( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))

Theorembj-opelidres 34892 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 34893 from it. (Contributed by BJ, 29-Mar-2020.)
(𝐴𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))

Theorembj-idreseq 34893 Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 34888 with V substituted for 𝑉 is a direct consequence of bj-idreseq 34893. This is a strengthening of resieq 5840 which should be proved from it (note that currently, resieq 5840 relies on ideq 5699). 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: ((𝐴𝐶𝐵𝐶) → ...). (Contributed by BJ, 25-Dec-2023.)
((𝐴𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵𝐴 = 𝐵))

Theorembj-idreseqb 34894 Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)
(𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 = 𝐵))

Theorembj-ideqg1 34895 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 34888 is in the main section.

((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))

Theorembj-ideqg1ALT 34896 Alternate proof of bj-ideqg1 using brabga 5396 instead of the "unbounded" version bj-brab2a1 34880 or brab2a 5619. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 34888 is in the main section.

((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))

Theorembj-opelidb1ALT 34897 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))

Theorembj-elid3 34898 Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.)
(⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

Theorembj-elid4 34899 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Theorembj-elid5 34900 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45831
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