Definition df-mre 16857

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 21686) and vector spaces (lssmre 19738) 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 16861, mresspw 16863, mre1cl 16865 and mreintcl 16866 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16871); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 16872. (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 16853	. 2 class Moore
2		vx	. . 3 setvar 𝑥
3		cvv 3494	. . 3 class V
4		vc	. . . . . 6 setvar 𝑐
5	2, 4	wel 2115	. . . . 5 wff 𝑥 ∈ 𝑐
6		vs	. . . . . . . . 9 setvar 𝑠
7	6	cv 1536	. . . . . . . 8 class 𝑠
8		c0 4291	. . . . . . . 8 class ∅
9	7, 8	wne 3016	. . . . . . 7 wff 𝑠 ≠ ∅
10	7	cint 4876	. . . . . . . 8 class ∩ 𝑠
11	4	cv 1536	. . . . . . . 8 class 𝑐
12	10, 11	wcel 2114	. . . . . . 7 wff ∩ 𝑠 ∈ 𝑐
13	9, 12	wi 4	. . . . . 6 wff (𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
14	11	cpw 4539	. . . . . 6 class 𝒫 𝑐
15	13, 6, 14	wral 3138	. . . . 5 wff ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐)
16	5, 15	wa 398	. . . 4 wff (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))
17	2	cv 1536	. . . . . 6 class 𝑥
18	17	cpw 4539	. . . . 5 class 𝒫 𝑥
19	18	cpw 4539	. . . 4 class 𝒫 𝒫 𝑥
20	16, 4, 19	crab 3142	. . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}
21	2, 3, 20	cmpt 5146	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
22	1, 21	wceq 1537	1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})

This definition is referenced by:  ismre  16861  fnmre  16862