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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 21686) and vector spaces (lssmre 19738)
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 16861, mresspw 16863, mre1cl 16865 and mreintcl 16866 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16871); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 16872. (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 16853 | . 2 class Moore | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3494 | . . 3 class V | |
4 | vc | . . . . . 6 setvar 𝑐 | |
5 | 2, 4 | wel 2115 | . . . . 5 wff 𝑥 ∈ 𝑐 |
6 | vs | . . . . . . . . 9 setvar 𝑠 | |
7 | 6 | cv 1536 | . . . . . . . 8 class 𝑠 |
8 | c0 4291 | . . . . . . . 8 class ∅ | |
9 | 7, 8 | wne 3016 | . . . . . . 7 wff 𝑠 ≠ ∅ |
10 | 7 | cint 4876 | . . . . . . . 8 class ∩ 𝑠 |
11 | 4 | cv 1536 | . . . . . . . 8 class 𝑐 |
12 | 10, 11 | wcel 2114 | . . . . . . 7 wff ∩ 𝑠 ∈ 𝑐 |
13 | 9, 12 | wi 4 | . . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐) |
14 | 11 | cpw 4539 | . . . . . 6 class 𝒫 𝑐 |
15 | 13, 6, 14 | wral 3138 | . . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐) |
16 | 5, 15 | wa 398 | . . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)) |
17 | 2 | cv 1536 | . . . . . 6 class 𝑥 |
18 | 17 | cpw 4539 | . . . . 5 class 𝒫 𝑥 |
19 | 18 | cpw 4539 | . . . 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 5146 | . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) |
22 | 1, 21 | wceq 1537 | 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 16861 fnmre 16862 |
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