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Definition df-mre 17629
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 23086) and vector spaces (lssmre 20964) 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 17633, mresspw 17635, mre1cl 17637 and mreintcl 17638 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17643); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 17644. (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 17625 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3480 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2109 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1539 . . . . . . . 8 class 𝑠
8 c0 4333 . . . . . . . 8 class
97, 8wne 2940 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4946 . . . . . . . 8 class 𝑠
114cv 1539 . . . . . . . 8 class 𝑐
1210, 11wcel 2108 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4600 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 3061 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 395 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1539 . . . . . 6 class 𝑥
1817cpw 4600 . . . . 5 class 𝒫 𝑥
1918cpw 4600 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 3436 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 5225 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1540 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  17633  fnmre  17634
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