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Definition df-mre 16847
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 21616) and vector spaces (lssmre 19669) 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 16851, mresspw 16853, mre1cl 16855 and mreintcl 16856 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16861); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16862. (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 16843 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3495 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2106 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1527 . . . . . . . 8 class 𝑠
8 c0 4290 . . . . . . . 8 class
97, 8wne 3016 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4869 . . . . . . . 8 class 𝑠
114cv 1527 . . . . . . . 8 class 𝑐
1210, 11wcel 2105 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4537 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 3138 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 396 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1527 . . . . . 6 class 𝑥
1817cpw 4537 . . . . 5 class 𝒫 𝑥
1918cpw 4537 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 3142 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 5138 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1528 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  16851  fnmre  16852
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