![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23026) and vector spaces (lssmre 20862)
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 17573, mresspw 17575, mre1cl 17577 and mreintcl 17578 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17583); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17584. (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 17565 | . 2 class Moore | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3461 | . . 3 class V | |
4 | vc | . . . . . 6 setvar 𝑐 | |
5 | 2, 4 | wel 2099 | . . . . 5 wff 𝑥 ∈ 𝑐 |
6 | vs | . . . . . . . . 9 setvar 𝑠 | |
7 | 6 | cv 1532 | . . . . . . . 8 class 𝑠 |
8 | c0 4322 | . . . . . . . 8 class ∅ | |
9 | 7, 8 | wne 2929 | . . . . . . 7 wff 𝑠 ≠ ∅ |
10 | 7 | cint 4950 | . . . . . . . 8 class ∩ 𝑠 |
11 | 4 | cv 1532 | . . . . . . . 8 class 𝑐 |
12 | 10, 11 | wcel 2098 | . . . . . . 7 wff ∩ 𝑠 ∈ 𝑐 |
13 | 9, 12 | wi 4 | . . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐) |
14 | 11 | cpw 4604 | . . . . . 6 class 𝒫 𝑐 |
15 | 13, 6, 14 | wral 3050 | . . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐) |
16 | 5, 15 | wa 394 | . . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)) |
17 | 2 | cv 1532 | . . . . . 6 class 𝑥 |
18 | 17 | cpw 4604 | . . . . 5 class 𝒫 𝑥 |
19 | 18 | cpw 4604 | . . . 4 class 𝒫 𝒫 𝑥 |
20 | 16, 4, 19 | crab 3418 | . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} |
21 | 2, 3, 20 | cmpt 5232 | . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) |
22 | 1, 21 | wceq 1533 | 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 17573 fnmre 17574 |
Copyright terms: Public domain | W3C validator |