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Definition df-mre 16017
Description: Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 20639) and vector spaces (lssmre 18735) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 16021, mresspw 16023, mre1cl 16025 and mreintcl 16026 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16031); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16032. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

Assertion
Ref Expression
df-mre Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
Distinct variable group:   𝑠,𝑐,𝑥

Detailed syntax breakdown of Definition df-mre
StepHypRef Expression
1 cmre 16013 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3172 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 1977 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1473 . . . . . . . 8 class 𝑠
8 c0 3873 . . . . . . . 8 class
97, 8wne 2779 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4404 . . . . . . . 8 class 𝑠
114cv 1473 . . . . . . . 8 class 𝑐
1210, 11wcel 1976 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4107 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 2895 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 382 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1473 . . . . . 6 class 𝑥
1817cpw 4107 . . . . 5 class 𝒫 𝑥
1918cpw 4107 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 2899 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 4637 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1474 1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
Colors of variables: wff setvar class
This definition is referenced by:  ismre  16021  fnmre  16022
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