Definition df-mre 17642

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 23244) and vector spaces (lssmre 21096) 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 17646, mresspw 17648, mre1cl 17650 and mreintcl 17651 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17656); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17657. (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 17638	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3455	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2144	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1569	. . . . . . . 8 class 𝑠
8		c0 4286	. . . . . . . 8 class ∅
9	7, 8	wne 2958	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4912	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1569	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2143	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4562	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3079	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 400	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1569	. . . . . 6 class 𝑥
18	17	cpw 4562	. . . . 5 class 𝒫 𝑥
19	18	cpw 4562	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 3416	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 5192	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
22	1, 21	wceq 1570	1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

This definition is used by:  ismre  17646  fnmre  17647