Definition df-mre 17756

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 23396) and vector spaces (lssmre 21241) 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 17760, mresspw 17762, mre1cl 17764 and mreintcl 17765 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17770); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17771. (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 17752	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3450	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2145	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1568	. . . . . . . 8 class 𝑠
8		c0 4278	. . . . . . . 8 class ∅
9	7, 8	wne 2955	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4906	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1568	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2144	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4556	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3076	. . . . 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 1568	. . . . . 6 class 𝑥
18	17	cpw 4556	. . . . 5 class 𝒫 𝑥
19	18	cpw 4556	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 3412	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 5185	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
22	1, 21	wceq 1569	1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

This definition is used by:  ismre  17760  fnmre  17761