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Theorem List for Metamath Proof Explorer - 34401-34500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-2uplth 34401 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5355). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theorembj-2uplex 34402 A couple is a set if and only if its coordinates are sets. For the advantages offered by the reverse closure property, see the section head comment. (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-2upln0 34403 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅

Theorembj-2upln1upl 34404 A couple is never equal to a monuple. It is in order to have this "non-clashing" result that tagging was used. Without tagging, we would have 𝐴, ∅⦆ = ⦅𝐴. Note that in the context of Morse tuples, it is natural to define the 0-tuple as the empty set. Therefore, the present theorem together with bj-1upln0 34389 and bj-2upln0 34403 tell us that an m-tuple may equal an n-tuple only when m = n, at least for m, n <= 2, but this result would extend as soon as we define n-tuples for higher values of n. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ⦅𝐶

20.15.5.16  Set theory: elementary operations relative to a universe

Some elementary set-theoretic operations "relative to a universe" (by which is merely meant some given class considered as a universe).

Theorembj-rcleqf 34405 Relative version of cleqf 3010. (Contributed by BJ, 27-Dec-2023.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑉       ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))

Theorembj-rcleq 34406* Relative version of dfcleq 2818. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))

Theorembj-reabeq 34407* Relative form of abeq2 2948. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))

Theorembj-disj2r 34408 Relative version of ssdifin0 4414, allowing a biconditional, and of disj2 4390. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4414 nor disj2 4390. (Proof modification is discouraged.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Theorembj-sscon 34409 Contraposition law for relative subclasses. Relative and generalized version of ssconb 4100, which it can shorten, as well as conss2 41067. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4100 nor conss2 41067. (Proof modification is discouraged.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

20.15.5.17  Set theory: miscellaneous

Miscellaneous theorems of set theory.

Theorembj-pw0ALT 34410 Alternate proof of pw0 4729. The proofs have a similar structure: pw0 4729 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 34410 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4729 and biconditional for bj-pw0ALT 34410) to translate the property ss0b 4334 into the wanted result. To translate a biconditional into a class equality, pw0 4729 uses abbii 2889 (which yields an equality of class abstractions), while bj-pw0ALT 34410 uses eqriv 2821 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2889, through its closed form abbi1 2887, is proved from eqrdv 2822, which is the deduction form of eqriv 2821. In the other direction, velpw 4527 and velsn 4566 are proved from the definitions of powerclass and singleton using elabg 3652, which is a version of abbii 2889 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 ∅ = {∅}

Theorembj-sselpwuni 34411 Quantitative version of ssexg 5213: a subset of an element of a class is an element of the powerclass of the union of that class. (Contributed by BJ, 6-Apr-2024.)
((𝐴𝐵𝐵𝑉) → 𝐴 ∈ 𝒫 𝑉)

Theorembj-unirel 34412 Quantitative version of uniexr 7479: if the union of a class is an element of a class, then that class is an element of the double powerclass of the union of this class. (Contributed by BJ, 6-Apr-2024.)
( 𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝑉)

Theorembj-elpwg 34413 If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4525 and elpw2g 5233 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theorembj-vjust 34414 Justification theorem for bj-df-v 34415. See also vjust 3481. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Theorembj-df-v 34415 Alternate definition of the universal class. Actually, the current definition df-v 3482 should be proved from this one, and vex 3483 should be proved from this proposed definition together with vexw 2808, which would remove from vex 3483 dependency on ax-13 2392 (see also comment of vexw 2808). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
V = {𝑥 ∣ ⊤}

Theorembj-df-nul 34416 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = {𝑥 ∣ ⊥}

Theorembj-nul 34417* Two formulations of the axiom of the empty set ax-nul 5196. Proposal: place it right before ax-nul 5196. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliota 34418* Definition of the empty set using the definite description binder. See also bj-nuliotaALT 34419. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliotaALT 34419* Alternate proof of bj-nuliota 34418. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6321). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-vtoclgfALT 34420 Alternate proof of vtoclgf 3551. Proof from vtoclgft 3539. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theorembj-elsn12g 34421 Join of elsng 4564 and elsn2g 4588. (Contributed by BJ, 18-Nov-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Theorembj-elsnb 34422 Biconditional version of elsng 4564. (Contributed by BJ, 18-Nov-2023.)
(𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Theorembj-pwcfsdom 34423 Remove hypothesis from pwcfsdom 10003. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 10003.) (Contributed by BJ, 14-Sep-2019.)
(ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))

Theorembj-grur1 34424 Remove hypothesis from grur1 10240. Illustration of how to remove a "definitional hypothesis". This makes its uses longer, but the theorem feels more self-contained. It looks preferable when the defined term appears only once in the conclusion. (Contributed by BJ, 14-Sep-2019.)
((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1‘(𝑈 ∩ On)))

Theorembj-bm1.3ii 34425* The extension of a predicate (𝜑(𝑧)) is included in a set (𝑥) if and only if it is a set (𝑦). Sufficiency is obvious, and necessity is the content of the axiom of separation ax-sep 5189. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.) Generalized to a closed form biconditional with existential quantifications using two different setvars 𝑥, 𝑦 (which need not be disjoint). (Revised by BJ, 8-Aug-2022.)

TODO: move in place of bm1.3ii 5192. Relabel ("sepbi"?).

(∃𝑥𝑧(𝜑𝑧𝑥) ↔ ∃𝑦𝑧(𝑧𝑦𝜑))

Theorembj-0nelopab 34426 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by BJ, 22-Jul-2023.)

TODO: move to the main section when one can reorder sections so that we can use relopab 5683 (this is a very limited reordering).

¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theorembj-brrelex12ALT 34427 Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5591. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-epelg 34428 The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5456 and closed form of epeli 5455. (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 5595 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
(𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

Theorembj-epelb 34429 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 5596 available. Check if it is shorter to prove bj-epelg 34428 first or bj-epelb 34429 first. (Contributed by BJ, 14-Jul-2023.)
(𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))

Theorembj-nsnid 34430 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 4638): ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
(𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)

20.15.5.18  Evaluation

This section treats the existing predicate Slot (df-slot 16487) as "evaluation at a class" and for the moment does not introduce new syntax for it.

Theorembj-evaleq 34431 Equality theorem for the Slot construction. This is currently a duplicate of sloteq 16488 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 34432 The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
Fun Slot 𝐴

Theorembj-evalfn 34433 The evaluation at a class is a function on the universal class. (General form of slotfn 16501). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
Slot 𝐴 Fn V

Theorembj-evalval 34434 Value of the evaluation at a class. (Closed form of strfvnd 16502 and strfvn 16505). (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 34435 The evaluation at a set of the identity function is that set. (General form of ndxarg 16508.) 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 34436 Proof of ndxarg 16508 from bj-evalid 34435. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theorembj-evalidval 34437 Closed general form of strndxid 16510. Both sides are equal to (𝐹𝐴) by bj-evalid 34435 and bj-evalval 34434 respectively, but bj-evalidval 34437 adds something to bj-evalid 34435 and bj-evalval 34434 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))

20.15.5.19  Elementwise operations

Syntaxcelwise 34438 Syntax for elementwise operations.
class elwise

Definitiondf-elwise 34439* Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 10565), 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)}))

20.15.5.20  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 16857), topologies (df-top 21502), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 31425), 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 34440 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 16703. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅

Theorembj-restsn 34441 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 34444 and bj-restsnid 34446. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Theorembj-restsnss 34442 Special case of bj-restsn 34441. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})

Theorembj-restsnss2 34443 Special case of bj-restsn 34441. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})

Theorembj-restsn0 34444 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 34441 and bj-restsnss2 34443. TODO: this is restsn 21778. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theorembj-restsn10 34445 Special case of bj-restsn 34441, bj-restsnss 34442, and bj-rest10 34447. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})

Theorembj-restsnid 34446 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 34441 and bj-restsnss 34442. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}

Theorembj-rest10 34447 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 21777 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))

Theorembj-rest10b 34448 Alternate version of bj-rest10 34447. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})

Theorembj-restn0 34449 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Theorembj-restn0b 34450 Alternate version of bj-restn0 34449. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)

Theorembj-restpw 34451 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 21786 (which uses distop 21603 and restopn2 21785). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Theorembj-rest0 34452 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 34453 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 34454 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 34455 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 34456 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 21770 and restuni2 21775. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Theorembj-restuni2 34457 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 21770 and restuni2 21775. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Theorembj-restreg 34458 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))

20.15.5.21  Moore collections (complements)

Theorembj-intss 34459 A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.)
(𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → 𝐴𝑋))

Theorembj-raldifsn 34460* 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 34461* 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 34462* A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 34464 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 34465.

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 7479).

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 5330) , 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 34471. (Contributed by BJ, 8-Dec-2021.)

(∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)

Syntaxcmoore 34463 Syntax for the class of Moore collections.
class Moore

Definitiondf-bj-moore 34464* 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 34462, and as illustrated by the lack of sethood condition in bj-ismoore 34465.

This is to df-mre 16857 (defining Moore) what df-top 21502 (defining Top) is to df-topon 21519 (defining TopOn).

For the sake of consistency, the function defined at df-mre 16857 should be denoted by "MooreOn".

Note: df-mre 16857 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 34461.

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 34462).

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 16857 and df-topon 21519) or the class of all families of that kind, independent of a base set (like df-bj-moore 34464 or df-top 21502). 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 34465* 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 34462 for the RHS). (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)

Theorembj-ismoored0 34466 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore 𝐴𝐴)

Theorembj-ismoored 34467 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)       (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)

Theorembj-ismoored2 34468 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ≠ ∅)       (𝜑 𝐵𝐴)

Theorembj-ismooredr 34469* 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 34470* Sufficient condition to be a Moore collection (variant of bj-ismooredr 34469 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 34471 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 34472 The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
¬ ∅ ∈ Moore

Theorembj-snmoore 34473 A singleton is a Moore collection. See bj-snmooreb 34474 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
(𝐴𝑉 → {𝐴} ∈ Moore)

Theorembj-snmooreb 34474 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 34475 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 34473). 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)

20.15.5.22  Maps-to notation for functions with three arguments

Theorembj-0nelmpt 34476 The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
¬ ∅ ∈ (𝑥𝐴𝐵)

Theorembj-mptval 34477 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 34478* An equivalent definition of df-mpo 7154. (Contributed by BJ, 30-Dec-2020.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Theorembj-mpomptALT 34479* Alternate proof of mpompt 7259. (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 34480 Syntax for maps-to notation for functions with three arguments.
class (𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷)

Definitiondf-bj-mpt3 34481* Define maps-to notation for functions with three arguments. See df-mpt 5133 and df-mpo 7154 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpoa 34478. (Contributed by BJ, 11-Apr-2020.)
(𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}

20.15.5.23  Currying

Currying and uncurrying. See also df-cur 7929 and df-unc 7930. Contrary to these, the definitions in this section are parameterized.

Syntaxcsethom 34482 Syntax for the set of set morphisms.
class Set

Definitiondf-bj-sethom 34483* Define the set of functions (morphisms of sets) between two sets. Same as df-map 8404 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 34484 Syntax for the set of topological morphisms.
class Top

Definitiondf-bj-tophom 34485* Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 21835 (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 34486 Syntax for the set of magma morphisms.
class Mgm

Definitiondf-bj-mgmhom 34487* 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 34488 Syntax for the set of topological magma morphisms.
class TopMgm

Definitiondf-bj-topmgmhom 34489* 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- 34490 Syntax for the parameterized currying function.
class curry_

Definitiondf-bj-cur 34491* Define currying. See also df-cur 7929. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Syntaxcunc- 34492 Notation for the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-unc 34493* Define uncurrying. See also df-unc 7930. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

20.15.5.24  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 34494 Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆

Definitiondf-strset 34495 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. (Contributed by BJ, 13-Feb-2022.)
[𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})

Theoremsetsstrset 34496 Relation between df-sets 16490 and df-strset 34495. 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 34593 and df-bj-rrhat 34595, and the point at infinity is denoted by , defined in df-bj-infty 34591.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 34589 (already defined as *, see df-xr 10677) and ℂ̅, defined in df-bj-ccbar 34576.

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 34497 Variant of nfald 2349. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theorembj-nfexd 34498 Variant of nfexd 2350. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremcopsex2d 34499* 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 34500* Biconditional form of copsex2d 34499. 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)))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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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 45201-45259
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