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Definition df-mre 16447
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 21093) and vector spaces (lssmre 19169) 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 16451, mresspw 16453, mre1cl 16455 and mreintcl 16456 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16461); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16462. (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 16443 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3391 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2157 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1636 . . . . . . . 8 class 𝑠
8 c0 4116 . . . . . . . 8 class
97, 8wne 2978 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4669 . . . . . . . 8 class 𝑠
114cv 1636 . . . . . . . 8 class 𝑐
1210, 11wcel 2156 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4351 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 3096 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 384 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1636 . . . . . 6 class 𝑥
1817cpw 4351 . . . . 5 class 𝒫 𝑥
1918cpw 4351 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 3100 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 4923 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1637 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  16451  fnmre  16452
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