Definition df-mre 17665

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 23290) and vector spaces (lssmre 21142) 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 17669, mresspw 17671, mre1cl 17673 and mreintcl 17674 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17679); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17680. (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 17661	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3457	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2147	. . . . 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 2960	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4914	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1569	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2146	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4564	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3081	. . . . 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 4564	. . . . 5 class 𝒫 𝑥
19	18	cpw 4564	. . . 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 5194	. 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  17669  fnmre  17670