Definition df-mre 17782

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 23423) and vector spaces (lssmre 21268) 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 17786, mresspw 17788, mre1cl 17790 and mreintcl 17791 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17796); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17797. (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 17778	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3454	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2148	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1569	. . . . . . . 8 class 𝑠
8		c0 4281	. . . . . . . 8 class ∅
9	7, 8	wne 2959	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4910	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1569	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2147	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4560	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3080	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 401	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1569	. . . . . 6 class 𝑥
18	17	cpw 4560	. . . . 5 class 𝒫 𝑥
19	18	cpw 4560	. . . 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 5189	. 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  17786  fnmre  17787