HomeHome Metamath Proof Explorer
Theorem List (p. 371 of 494)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30937)
  Hilbert Space Explorer  Hilbert Space Explorer
(30938-32460)
  Users' Mathboxes  Users' Mathboxes
(32461-49324)
 

Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxbj-cpr1 37001 Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴
 
Definitiondf-bj-pr1 37002 Definition of the first projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr1eq 37003, bj-pr11val 37006, bj-pr21val 37014, bj-pr1ex 37007. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
pr1 𝐴 = (∅ Proj 𝐴)
 
Theorembj-pr1eq 37003 Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
 
Theorembj-pr1un 37004 The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
 
Theorembj-pr1val 37005 Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
 
Theorembj-pr11val 37006 Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
pr1𝐴⦆ = 𝐴
 
Theorembj-pr1ex 37007 Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr1 𝐴 ∈ V)
 
Theorembj-1uplth 37008 The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
 
Theorembj-1uplex 37009 A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)
 
Theorembj-1upln0 37010 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
𝐴⦆ ≠ ∅
 
Syntaxbj-c2uple 37011 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class 𝐴, 𝐵
 
Definitiondf-bj-2upl 37012 Definition of the Morse couple. See df-bj-1upl 36999. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 37013, bj-2uplth 37022, bj-2uplex 37023, and the properties of the projections (see df-bj-pr1 37002 and df-bj-pr2 37016). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
 
Theorembj-2upleq 37013 Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
 
Theorembj-pr21val 37014 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr1𝐴, 𝐵⦆ = 𝐴
 
Syntaxbj-cpr2 37015 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴
 
Definitiondf-bj-pr2 37016 Definition of the second projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr2eq 37017, bj-pr22val 37020, bj-pr2ex 37021. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
pr2 𝐴 = (1o Proj 𝐴)
 
Theorembj-pr2eq 37017 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
 
Theorembj-pr2un 37018 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
 
Theorembj-pr2val 37019 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
 
Theorembj-pr22val 37020 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr2𝐴, 𝐵⦆ = 𝐵
 
Theorembj-pr2ex 37021 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr2 𝐴 ∈ V)
 
Theorembj-2uplth 37022 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5481). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theorembj-2uplex 37023 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 37024 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅
 
Theorembj-2upln1upl 37025 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 37010 and bj-2upln0 37024 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
 
21.19.5.16  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 37026 Relative version of cleqf 2934. (Contributed by BJ, 27-Dec-2023.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑉       ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
 
Theorembj-rcleq 37027* Relative version of dfcleq 2730. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
 
Theorembj-reabeq 37028* Relative form of eqabb 2881. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
 
Theorembj-disj2r 37029 Relative version of ssdifin0 4486, allowing a biconditional, and of disj2 4458. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4486 nor disj2 4458. (Proof modification is discouraged.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
 
Theorembj-sscon 37030 Contraposition law for relative subclasses. Relative and generalized version of ssconb 4142, which it can shorten, as well as conss2 44462. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4142 nor conss2 44462. (Proof modification is discouraged.)
((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
 
21.19.5.17  Axioms for finite unions

In this section, we introduce the axiom of singleton ax-bj-sn 37034 and the axiom of binary union ax-bj-bun 37038. Both axioms are implied by the standard axioms of unordered pair ax-pr 5432 and of union ax-un 7755 (see snex 5436 and unex 7764). Conversely, the axiom of unordered pair ax-pr 5432 is implied by the axioms of singleton and of binary union, as proved in bj-prexg 37040 and bj-prex 37041.

The axioms of union ax-un 7755 and of powerset ax-pow 5365 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 5365 and https://mathoverflow.net/questions/48365 5365.

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 5306).

The axiom of binary union is useful in theories without the axioms of union ax-un 7755 and of powerset ax-pow 5365. 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 2708, ax-rep 5279, ax-sep 5296, ax-nul 5306, ax-reg 9632 } and the axioms of existence of unordered 𝑚-tuples for all 𝑚𝑛, 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 37043 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 37047 and conversely how to prove from adjunction singleton (bj-snfromadj 37045) and unordered pair (bj-prfromadj 37046).

 
Theorembj-abex 37031* Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
({𝑥𝜑} ∈ V ↔ ∃𝑦𝑥(𝑥𝑦𝜑))
 
Theorembj-clex 37032* Two ways of stating that a class is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
(𝑥𝐴𝜑)       (𝐴 ∈ V ↔ ∃𝑦𝑥(𝑥𝑦𝜑))
 
Theorembj-axsn 37033* Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37034). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
({𝑥} ∈ V ↔ ∃𝑦𝑧(𝑧𝑦𝑧 = 𝑥))
 
Axiomax-bj-sn 37034* Axiom of singleton. (Contributed by BJ, 12-Jan-2025.)
𝑥𝑦𝑧(𝑧𝑦𝑧 = 𝑥)
 
Theorembj-snexg 37035 A singleton built on a set is a set. Contrary to bj-snex 37036, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5436 and prove it from ax-bj-sn 37034. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)
 
Theorembj-snex 37036 A singleton is a set. See also snex 5436, snexALT 5383. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37034. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
{𝐴} ∈ V
 
Theorembj-axbun 37037* Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 37038). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝑥𝑦) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦)))
 
Axiomax-bj-bun 37038* Axiom of binary union. (Contributed by BJ, 12-Jan-2025.)
𝑥𝑦𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
 
Theorembj-unexg 37039 Existence of binary unions of sets, proved from ax-bj-bun 37038. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theorembj-prexg 37040 Existence of unordered pairs formed on sets, proved from ax-bj-sn 37034 and ax-bj-bun 37038. Contrary to bj-prex 37041, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
 
Theorembj-prex 37041 Existence of unordered pairs proved from ax-bj-sn 37034 and ax-bj-bun 37038. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
{𝐴, 𝐵} ∈ V
 
Theorembj-axadj 37042* Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37043). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡 = 𝑦)))
 
Axiomax-bj-adj 37043* Axiom of adjunction. (Contributed by BJ, 19-Jan-2025.)
𝑥𝑦𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡 = 𝑦))
 
Theorembj-adjg1 37044 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.)
(𝐴𝑉 → (𝐴 ∪ {𝑥}) ∈ V)
 
Theorembj-snfromadj 37045 Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
{𝑥} ∈ V
 
Theorembj-prfromadj 37046 Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V
 
Theorembj-adjfrombun 37047 Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
(𝑥 ∪ {𝑦}) ∈ V
 
21.19.5.18  Set theory: miscellaneous

Miscellaneous theorems of set theory.

 
Theoremeleq2w2ALT 37048 Alternate proof of eleq2w2 2733 and special instance of eleq2 2830. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
 
Theorembj-clel3gALT 37049* Alternate proof of clel3g 3661. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
 
Theorembj-pw0ALT 37050 Alternate proof of pw0 4812. The proofs have a similar structure: pw0 4812 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 37050 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4812 and biconditional for bj-pw0ALT 37050) to translate the property ss0b 4401 into the wanted result. To translate a biconditional into a class equality, pw0 4812 uses abbii 2809 (which yields an equality of class abstractions), while bj-pw0ALT 37050 uses eqriv 2734 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2809, through its closed form abbi 2807, is proved from eqrdv 2735, which is the deduction form of eqriv 2734. In the other direction, velpw 4605 and velsn 4642 are proved from the definitions of powerclass and singleton using elabg 3676, which is a version of abbii 2809 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 ∅ = {∅}
 
Theorembj-sselpwuni 37051 Quantitative version of ssexg 5323: 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 37052 Quantitative version of uniexr 7783: 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 37053 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 4603 and elpw2g 5333 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theorembj-velpwALT 37054* This theorem bj-velpwALT 37054 and the next theorem bj-elpwgALT 37055 are alternate proofs of velpw 4605 and elpwg 4603 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3557 instead of proving first the general case using elab2g 3680 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2177. In other cases, that order is better (e.g., vsnex 5434 proved before snexg 5435). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)
 
Theorembj-elpwgALT 37055 Alternate proof of elpwg 4603. See comment for bj-velpwALT 37054. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theorembj-vjust 37056 Justification theorem for dfv2 3483 if it were the definition. See also vjust 3481. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
 
Theorembj-nul 37057* Two formulations of the axiom of the empty set ax-nul 5306. Proposal: place it right before ax-nul 5306. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
 
Theorembj-nuliota 37058* Definition of the empty set using the definite description binder. See also bj-nuliotaALT 37059. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
 
Theorembj-nuliotaALT 37059* Alternate proof of bj-nuliota 37058. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6539). 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 37060 Alternate proof of vtoclgf 3569. Proof from vtoclgft 3552. (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 37061 Join of elsng 4640 and elsn2g 4664. (Contributed by BJ, 18-Nov-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theorembj-elsnb 37062 Biconditional version of elsng 4640. (Contributed by BJ, 18-Nov-2023.)
(𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
Theorembj-pwcfsdom 37063 Remove hypothesis from pwcfsdom 10623. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 10623.) (Contributed by BJ, 14-Sep-2019.)
(ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
 
Theorembj-grur1 37064 Remove hypothesis from grur1 10860. 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 37065* 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 5296. 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 after sepexi 5301. Relabel ("sepbi"?).

(∃𝑥𝑧(𝜑𝑧𝑥) ↔ ∃𝑦𝑧(𝑧𝑦𝜑))
 
Theorembj-dfid2ALT 37066 Alternate version of dfid2 5580. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5578 instead to make the semantics of the construction df-opab 5206 clearer. (New usage is discouraged.)
I = {⟨𝑥, 𝑥⟩ ∣ ⊤}
 
Theorembj-0nelopab 37067 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 5834 (this is a very limited reordering).

¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theorembj-brrelex12ALT 37068 Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5737. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorembj-epelg 37069 The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5587 and closed form of epeli 5586. (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 5741 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
(𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
 
Theorembj-epelb 37070 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 5742 available. Check if it is shorter to prove bj-epelg 37069 first or bj-epelb 37070 first. (Contributed by BJ, 14-Jul-2023.)
(𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))
 
Theorembj-nsnid 37071 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 4717): ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
(𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)
 
Theorembj-rdg0gALT 37072 Alternate proof of rdg0g 8467. More direct since it bypasses tz7.44-1 8446 and rdg0 8461 (and vtoclg 3554, vtoclga 3577). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
 
21.19.5.19  Evaluation at a class

This section treats the existing predicate Slot (df-slot 17219) as "evaluation at a class" and for the moment does not introduce new syntax for it.

 
Theorembj-evaleq 37073 Equality theorem for the Slot construction. This is currently a duplicate of sloteq 17220 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 37074 The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.)
Fun Slot 𝐴
 
Theorembj-evalfn 37075 The evaluation at a class is a function on the universal class. (General form of slotfn 17221). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
Slot 𝐴 Fn V
 
Theorembj-evalval 37076 Value of the evaluation at a class. (Closed form of strfvnd 17222 and strfvn 17223). (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 37077 The evaluation at a set of the identity function is that set. (General form of ndxarg 17233.) 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 37078 Proof of ndxarg 17233 from bj-evalid 37077. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁
 
Theorembj-evalidval 37079 Closed general form of strndxid 17235. Both sides are equal to (𝐹𝐴) by bj-evalid 37077 and bj-evalval 37076 respectively, but bj-evalidval 37079 adds something to bj-evalid 37077 and bj-evalval 37076 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))
 
21.19.5.20  Elementwise operations
 
Syntaxcelwise 37080 Syntax for elementwise operations.
class elwise
 
Definitiondf-elwise 37081* Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 11185), 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)}))
 
21.19.5.21  Elementwise intersection (families of sets induced on a subset)

Many kinds of structures are given by families of subsets of a given set: Moore collections (df-mre 17629), topologies (df-top 22900), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 34110), 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 37082 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 17474. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅
 
Theorembj-restsn 37083 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 37086 and bj-restsnid 37088. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})
 
Theorembj-restsnss 37084 Special case of bj-restsn 37083. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})
 
Theorembj-restsnss2 37085 Special case of bj-restsn 37083. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})
 
Theorembj-restsn0 37086 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 37083 and bj-restsnss2 37085. TODO: this is restsn 23178. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})
 
Theorembj-restsn10 37087 Special case of bj-restsn 37083, bj-restsnss 37084, and bj-rest10 37089. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})
 
Theorembj-restsnid 37088 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 37083 and bj-restsnss 37084. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}
 
Theorembj-rest10 37089 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 23177 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))
 
Theorembj-rest10b 37090 Alternate version of bj-rest10 37089. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})
 
Theorembj-restn0 37091 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))
 
Theorembj-restn0b 37092 Alternate version of bj-restn0 37091. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)
 
Theorembj-restpw 37093 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 23186 (which uses distop 23002 and restopn2 23185). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))
 
Theorembj-rest0 37094 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 37095 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 37096 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 37097 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 37098 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 23170 and restuni2 23175. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))
 
Theorembj-restuni2 37099 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23170 and restuni2 23175. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)
 
Theorembj-restreg 37100 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))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 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 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49324
  Copyright terms: Public domain < Previous  Next >