Definition df-mre 16017

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 20639) and vector spaces (lssmre 18735) 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 16021, mresspw 16023, mre1cl 16025 and mreintcl 16026 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16031); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 16032. (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 16013	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3172	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 1977	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1473	. . . . . . . 8 class 𝑠
8		c0 3873	. . . . . . . 8 class ∅
9	7, 8	wne 2779	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4404	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1473	. . . . . . . 8 class 𝑐
12	10, 11	wcel 1976	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4107	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 2895	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 382	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1473	. . . . . 6 class 𝑥
18	17	cpw 4107	. . . . 5 class 𝒫 𝑥
19	18	cpw 4107	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 2899	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 4637	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
22	1, 21	wceq 1474	1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

This definition is referenced by:  ismre  16021  fnmre  16022