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Definition df-mre 17630
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 23101) and vector spaces (lssmre 20981) 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 17634, mresspw 17636, mre1cl 17638 and mreintcl 17639 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17644); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 17645. (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 17626 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3477 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2106 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1535 . . . . . . . 8 class 𝑠
8 c0 4338 . . . . . . . 8 class
97, 8wne 2937 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4950 . . . . . . . 8 class 𝑠
114cv 1535 . . . . . . . 8 class 𝑐
1210, 11wcel 2105 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4604 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 3058 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 395 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1535 . . . . . 6 class 𝑥
1817cpw 4604 . . . . 5 class 𝒫 𝑥
1918cpw 4604 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 3432 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 5230 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1536 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  17634  fnmre  17635
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