Definition df-mre 17530

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 22582) and vector spaces (lssmre 20577) 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 17534, mresspw 17536, mre1cl 17538 and mreintcl 17539 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17544); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17545. (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 17526	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3475	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2108	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1541	. . . . . . . 8 class 𝑠
8		c0 4323	. . . . . . . 8 class ∅
9	7, 8	wne 2941	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4951	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1541	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2107	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4603	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3062	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 397	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1541	. . . . . 6 class 𝑥
18	17	cpw 4603	. . . . 5 class 𝒫 𝑥
19	18	cpw 4603	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 3433	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 5232	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
22	1, 21	wceq 1542	1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

This definition is referenced by:  ismre  17534  fnmre  17535