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Mirrors > Home > MPE Home > Th. List > df-mre | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-mre | ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmre 16843 | . 2 class Moore | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3495 | . . 3 class V | |
4 | vc | . . . . . 6 setvar 𝑐 | |
5 | 2, 4 | wel 2106 | . . . . 5 wff 𝑥 ∈ 𝑐 |
6 | vs | . . . . . . . . 9 setvar 𝑠 | |
7 | 6 | cv 1527 | . . . . . . . 8 class 𝑠 |
8 | c0 4290 | . . . . . . . 8 class ∅ | |
9 | 7, 8 | wne 3016 | . . . . . . 7 wff 𝑠 ≠ ∅ |
10 | 7 | cint 4869 | . . . . . . . 8 class ∩ 𝑠 |
11 | 4 | cv 1527 | . . . . . . . 8 class 𝑐 |
12 | 10, 11 | wcel 2105 | . . . . . . 7 wff ∩ 𝑠 ∈ 𝑐 |
13 | 9, 12 | wi 4 | . . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐) |
14 | 11 | cpw 4537 | . . . . . 6 class 𝒫 𝑐 |
15 | 13, 6, 14 | wral 3138 | . . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐) |
16 | 5, 15 | wa 396 | . . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)) |
17 | 2 | cv 1527 | . . . . . 6 class 𝑥 |
18 | 17 | cpw 4537 | . . . . 5 class 𝒫 𝑥 |
19 | 18 | cpw 4537 | . . . 4 class 𝒫 𝒫 𝑥 |
20 | 16, 4, 19 | crab 3142 | . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} |
21 | 2, 3, 20 | cmpt 5138 | . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) |
22 | 1, 21 | wceq 1528 | 1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) |
Colors of variables: wff setvar class |
This definition is referenced by: ismre 16851 fnmre 16852 |
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