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Definition df-mre 17569
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 23026) and vector spaces (lssmre 20862) 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 17573, mresspw 17575, mre1cl 17577 and mreintcl 17578 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17583); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 17584. (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 17565 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3461 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2099 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1532 . . . . . . . 8 class 𝑠
8 c0 4322 . . . . . . . 8 class
97, 8wne 2929 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4950 . . . . . . . 8 class 𝑠
114cv 1532 . . . . . . . 8 class 𝑐
1210, 11wcel 2098 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4604 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 3050 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 394 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1532 . . . . . 6 class 𝑥
1817cpw 4604 . . . . 5 class 𝒫 𝑥
1918cpw 4604 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 3418 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 5232 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1533 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  17573  fnmre  17574
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