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Definition df-mre 16293
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 20930) and vector spaces (lssmre 19014) 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 16297, mresspw 16299, mre1cl 16301 and mreintcl 16302 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16307); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16308. (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 16289 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3231 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2031 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1522 . . . . . . . 8 class 𝑠
8 c0 3948 . . . . . . . 8 class
97, 8wne 2823 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4507 . . . . . . . 8 class 𝑠
114cv 1522 . . . . . . . 8 class 𝑐
1210, 11wcel 2030 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4191 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 2941 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 383 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1522 . . . . . 6 class 𝑥
1817cpw 4191 . . . . 5 class 𝒫 𝑥
1918cpw 4191 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 2945 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 4762 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1523 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  16297  fnmre  16298
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