Definition df-mre 17503

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 23020) and vector spaces (lssmre 20915) 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 17507, mresspw 17509, mre1cl 17511 and mreintcl 17512 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17517); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17518. (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 17499	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3438	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2114	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1540	. . . . . . . 8 class 𝑠
8		c0 4283	. . . . . . . 8 class ∅
9	7, 8	wne 2930	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4900	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1540	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2113	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4552	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3049	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 395	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1540	. . . . . 6 class 𝑥
18	17	cpw 4552	. . . . 5 class 𝒫 𝑥
19	18	cpw 4552	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 3397	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 5177	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
22	1, 21	wceq 1541	1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

This definition is referenced by:  ismre  17507  fnmre  17508