Definition df-mre 17644

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 23107) and vector spaces (lssmre 20987) 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 17648, mresspw 17650, mre1cl 17652 and mreintcl 17653 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17658); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17659. (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 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

Distinct variable group:   𝑠,𝑐,𝑥

Detailed syntax breakdown of Definition df-mre

Step	Hyp	Ref	Expression
1		cmre 17640	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3488	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2109	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1536	. . . . . . . 8 class 𝑠
8		c0 4352	. . . . . . . 8 class ∅
9	7, 8	wne 2946	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4970	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1536	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2108	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4622	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3067	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 395	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1536	. . . . . 6 class 𝑥
18	17	cpw 4622	. . . . 5 class 𝒫 𝑥
19	18	cpw 4622	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 3443	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 5249	. 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))})

This definition is referenced by:  ismre  17648  fnmre  17649