P. 358 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-pr2eq 35701	Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
Theorem	bj-pr2un 35702	The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Theorem	bj-pr2val 35703	Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Theorem	bj-pr22val 35704	Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
Theorem	bj-pr2ex 35705	Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V)
Theorem	bj-2uplth 35706	The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5469). (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	bj-2uplex 35707	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))
Theorem	bj-2upln0 35708	A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ∅
Theorem	bj-2upln1upl 35709	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 35694 and bj-2upln0 35708 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⦆
21.15.5.16  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).
Theorem	bj-rcleqf 35710	Relative version of cleqf 2933. (Contributed by BJ, 27-Dec-2023.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑉    ⇒   ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-rcleq 35711*	Relative version of dfcleq 2724. (Contributed by BJ, 27-Dec-2023.)
⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-reabeq 35712*	Relative form of eqabb 2872. (Contributed by BJ, 27-Dec-2023.)
⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	bj-disj2r 35713	Relative version of ssdifin0 4479, allowing a biconditional, and of disj2 4453. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4479 nor disj2 4453. (Proof modification is discouraged.)
⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)
Theorem	bj-sscon 35714	Contraposition law for relative subclasses. Relative and generalized version of ssconb 4133, which it can shorten, as well as conss2 42973. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4133 nor conss2 42973. (Proof modification is discouraged.)
⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴))
21.15.5.17  Axioms for finite unions In this section, we introduce the axiom of singleton ax-bj-sn 35718 and the axiom of binary union ax-bj-bun 35722. Both axioms are implied by the standard axioms of unordered pair ax-pr 5420 and of union ax-un 7708 (see snex 5424 and unex 7716). Conversely, the axiom of unordered pair ax-pr 5420 is implied by the axioms of singleton and of binary union, as proved in bj-prexg 35724 and bj-prex 35725. The axioms of union ax-un 7708 and of powerset ax-pow 5356 are independent of these axioms: consider respectively the class of pseudo-hereditarily sets of cardinality less than a given singular strong limit cardinal, see Greg Oman, On the axiom of union, Arch. Math. Logic (2010) 49:283--289 (that model does have finite unions), and the class of well-founded hereditarily countable sets (or hereditarily less than a given uncountable regular cardinal). See also https://mathoverflow.net/questions/81815 5356 and https://mathoverflow.net/questions/48365 5356. A proof by finite induction shows that the existence of finite unions is equivalent to the existence of binary unions and of nullary unions (the latter being the axiom of the empty set ax-nul 5299). The axiom of binary union is useful in theories without the axioms of union ax-un 7708 and of powerset ax-pow 5356. For instance, the class of well-founded sets hereditarily of cardinality at most 𝑛 ∈ ℕ0 with ordinary membership relation is a model of { ax-ext 2702, ax-rep 5278, ax-sep 5292, ax-nul 5299, ax-reg 9569 } and the axioms of existence of unordered 𝑚-tuples for all 𝑚 ≤ 𝑛, and in most cases one would like to rule out such models, hence the need for extra axioms, typically variants of powersets or unions. The axiom of adjunction ax-bj-adj 35727 is more widely used, and is an axiom of General Set Theory. We prove how to retrieve it from binary union and singleton in bj-adjfrombun 35731 and conversely how to prove from adjunction singleton (bj-snfromadj 35729) and unordered pair (bj-prfromadj 35730).
Theorem	bj-abex 35715*	Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
Theorem	bj-clex 35716*	Two ways of stating that a class is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)    ⇒   ⊢ (𝐴 ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
Theorem	bj-axsn 35717*	Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 35718). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
Axiom	ax-bj-sn 35718*	Axiom of singleton. (Contributed by BJ, 12-Jan-2025.)
⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
Theorem	bj-snexg 35719	A singleton built on a set is a set. Contrary to bj-snex 35720, this proof is intuitionistically valid and does not require ax-nul 5299. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5424 and prove it from ax-bj-sn 35718. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 35720	A singleton is a set. See also snex 5424, snexALT 5374. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 35718. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝐴} ∈ V
Theorem	bj-axbun 35721*	Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 35722). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))
Axiom	ax-bj-bun 35722*	Axiom of binary union. (Contributed by BJ, 12-Jan-2025.)
⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
Theorem	bj-unexg 35723	Existence of binary unions of sets, proved from ax-bj-bun 35722. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-prexg 35724	Existence of unordered pairs formed on sets, proved from ax-bj-sn 35718 and ax-bj-bun 35722. Contrary to bj-prex 35725, this proof is intuitionistically valid and does not require ax-nul 5299. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-prex 35725	Existence of unordered pairs proved from ax-bj-sn 35718 and ax-bj-bun 35722. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝐴, 𝐵} ∈ V
Theorem	bj-axadj 35726*	Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 35727). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))
Axiom	ax-bj-adj 35727*	Axiom of adjunction. (Contributed by BJ, 19-Jan-2025.)
⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
Theorem	bj-adjg1 35728	Existence of the result of the adjunction (generalized only in the first term since this suffices for current applications). (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)
Theorem	bj-snfromadj 35729	Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝑥} ∈ V
Theorem	bj-prfromadj 35730	Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-adjfrombun 35731	Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝑥 ∪ {𝑦}) ∈ V
21.15.5.18  Set theory: miscellaneous Miscellaneous theorems of set theory.
Theorem	eleq2w2ALT 35732	Alternate proof of eleq2w2 2727 and special instance of eleq2 2821. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	bj-clel3gALT 35733*	Alternate proof of clel3g 3646. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
Theorem	bj-pw0ALT 35734	Alternate proof of pw0 4808. The proofs have a similar structure: pw0 4808 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 35734 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4808 and biconditional for bj-pw0ALT 35734) to translate the property ss0b 4393 into the wanted result. To translate a biconditional into a class equality, pw0 4808 uses abbii 2801 (which yields an equality of class abstractions), while bj-pw0ALT 35734 uses eqriv 2728 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2801, through its closed form abbi 2799, is proved from eqrdv 2729, which is the deduction form of eqriv 2728. In the other direction, velpw 4601 and velsn 4638 are proved from the definitions of powerclass and singleton using elabg 3662, which is a version of abbii 2801 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝒫 ∅ = {∅}
Theorem	bj-sselpwuni 35735	Quantitative version of ssexg 5316: 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.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)
Theorem	bj-unirel 35736	Quantitative version of uniexr 7733: 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.)
⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)
Theorem	bj-elpwg 35737	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 4599 and elpw2g 5337 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	bj-velpwALT 35738*	This theorem bj-velpwALT 35738 and the next theorem bj-elpwgALT 35739 are alternate proofs of velpw 4601 and elpwg 4599 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3556 instead of proving first the general case using elab2g 3666 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2171. In other cases, that order is better (e.g., vsnex 5422 proved before snexg 5423). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
Theorem	bj-elpwgALT 35739	Alternate proof of elpwg 4599. See comment for bj-velpwALT 35738. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	bj-vjust 35740	Justification theorem for dfv2 3476 if it were the definition. See also vjust 3474. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
Theorem	bj-nul 35741*	Two formulations of the axiom of the empty set ax-nul 5299. Proposal: place it right before ax-nul 5299. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliota 35742*	Definition of the empty set using the definite description binder. See also bj-nuliotaALT 35743. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliotaALT 35743*	Alternate proof of bj-nuliota 35742. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6510). 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.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-vtoclgfALT 35744	Alternate proof of vtoclgf 3551. Proof from vtoclgft 3537. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-elsn12g 35745	Join of elsng 4636 and elsn2g 4660. (Contributed by BJ, 18-Nov-2023.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	bj-elsnb 35746	Biconditional version of elsng 4636. (Contributed by BJ, 18-Nov-2023.)
⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-pwcfsdom 35747	Remove hypothesis from pwcfsdom 10560. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 10560.) (Contributed by BJ, 14-Sep-2019.)
⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
Theorem	bj-grur1 35748	Remove hypothesis from grur1 10797. 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)))
Theorem	bj-bm1.3ii 35749*	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 5292. 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 5295. Relabel ("sepbi"?).
⊢ (∃𝑥∀𝑧(𝜑 → 𝑧 ∈ 𝑥) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝜑))
Theorem	bj-dfid2ALT 35750	Alternate version of dfid2 5569. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5567 instead to make the semantics of the construction df-opab 5204 clearer. (New usage is discouraged.)
⊢ I = {⟨𝑥, 𝑥⟩ ∣ ⊤}
Theorem	bj-0nelopab 35751	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 5816 (this is a very limited reordering).
⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	bj-brrelex12ALT 35752	Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5720. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-epelg 35753	The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5576 and closed form of epeli 5575. (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 5724 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	bj-epelb 35754	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 5725 available. Check if it is shorter to prove bj-epelg 35753 first or bj-epelb 35754 first. (Contributed by BJ, 14-Jul-2023.)
⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
Theorem	bj-nsnid 35755	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 4714): ⊢ ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴 → 𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ∈ 𝐴)
Theorem	bj-rdg0gALT 35756	Alternate proof of rdg0g 8409. More direct since it bypasses tz7.44-1 8388 and rdg0 8403 (and vtoclg 3553, vtoclga 3562). (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)
21.15.5.19  Evaluation at a class This section treats the existing predicate Slot (df-slot 17097) as "evaluation at a class" and for the moment does not introduce new syntax for it.
Theorem	bj-evaleq 35757	Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17098 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 35758	The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
⊢ Fun Slot 𝐴
Theorem	bj-evalfn 35759	The evaluation at a class is a function on the universal class. (General form of slotfn 17099). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
⊢ Slot 𝐴 Fn V
Theorem	bj-evalval 35760	Value of the evaluation at a class. (Closed form of strfvnd 17100 and strfvn 17101). (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 35761	The evaluation at a set of the identity function is that set. (General form of ndxarg 17111.) 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 35762	Proof of ndxarg 17111 from bj-evalid 35761. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁
Theorem	bj-evalidval 35763	Closed general form of strndxid 17113. Both sides are equal to (𝐹‘𝐴) by bj-evalid 35761 and bj-evalval 35760 respectively, but bj-evalidval 35763 adds something to bj-evalid 35761 and bj-evalval 35760 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))
21.15.5.20  Elementwise operations
Syntax	celwise 35764	Syntax for elementwise operations.
class elwise
Definition	df-elwise 35765*	Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 11122), 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)}))
21.15.5.21  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 17512), topologies (df-top 22325), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 32938), 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 35766	An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 17357. (Contributed by BJ, 27-Apr-2021.)
⊢ (∅ ↾t 𝐴) = ∅
Theorem	bj-restsn 35767	An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 35770 and bj-restsnid 35772. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})
Theorem	bj-restsnss 35768	Special case of bj-restsn 35767. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})
Theorem	bj-restsnss2 35769	Special case of bj-restsn 35767. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})
Theorem	bj-restsn0 35770	An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 35767 and bj-restsnss2 35769. TODO: this is restsn 22603. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅})
Theorem	bj-restsn10 35771	Special case of bj-restsn 35767, bj-restsnss 35768, and bj-rest10 35773. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → ({𝑋} ↾t ∅) = {∅})
Theorem	bj-restsnid 35772	The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 35767 and bj-restsnss 35768. (Contributed by BJ, 27-Apr-2021.)
⊢ ({𝐴} ↾t 𝐴) = {𝐴}
Theorem	bj-rest10 35773	An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 22602 and could replace it. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅}))
Theorem	bj-rest10b 35774	Alternate version of bj-rest10 35773. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅})
Theorem	bj-restn0 35775	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 35776	Alternate version of bj-restn0 35775. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴 ∈ 𝑊) → (𝑋 ↾t 𝐴) ≠ ∅)
Theorem	bj-restpw 35777	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 22611 (which uses distop 22427 and restopn2 22610). (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))
Theorem	bj-rest0 35778	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 35779	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 35780	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 35781	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 35782	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 22595 and restuni2 22600. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))
Theorem	bj-restuni2 35783	The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 22595 and restuni2 22600. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)
Theorem	bj-restreg 35784	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))
21.15.5.22  Moore collections (complements)
Theorem	bj-raldifsn 35785*	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 35786*	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 35787*	A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 35789 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 35790. 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 7733). 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 5443), 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 35796. (Contributed by BJ, 8-Dec-2021.)
⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
Syntax	cmoore 35788	Syntax for the class of Moore collections.
class Moore
Definition	df-bj-moore 35789*	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 35787, and as illustrated by the lack of sethood condition in bj-ismoore 35790. This is to df-mre 17512 (defining Moore) what df-top 22325 (defining Top) is to df-topon 22342 (defining TopOn). For the sake of consistency, the function defined at df-mre 17512 should be denoted by "MooreOn". Note: df-mre 17512 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 35786. 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 35787). 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 17512 and df-topon 22342) or the class of all families of that kind, independent of a base set (like df-bj-moore 35789 or df-top 22325). 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 35790*	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 35787 for the RHS). (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
Theorem	bj-ismoored0 35791	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)
Theorem	bj-ismoored 35792	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 35793	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 35794*	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 35795*	Sufficient condition to be a Moore collection (variant of bj-ismooredr 35794 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 35796	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 35797	The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ ¬ ∅ ∈ Moore
Theorem	bj-snmoore 35798	A singleton is a Moore collection. See bj-snmooreb 35799 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)
Theorem	bj-snmooreb 35799	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 35800	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 35798). 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)