MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mre Structured version   Visualization version   GIF version

Definition df-mre 16686
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 21370) and vector spaces (lssmre 19428) 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 16690, mresspw 16692, mre1cl 16694 and mreintcl 16695 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16700); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16701. (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 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
Distinct variable group:   𝑠,𝑐,𝑥

Detailed syntax breakdown of Definition df-mre
StepHypRef Expression
1 cmre 16682 . 2 class Moore
2 vx . . 3 setvar 𝑥
3 cvv 3437 . . 3 class V
4 vc . . . . . 6 setvar 𝑐
52, 4wel 2082 . . . . 5 wff 𝑥𝑐
6 vs . . . . . . . . 9 setvar 𝑠
76cv 1521 . . . . . . . 8 class 𝑠
8 c0 4211 . . . . . . . 8 class
97, 8wne 2984 . . . . . . 7 wff 𝑠 ≠ ∅
107cint 4782 . . . . . . . 8 class 𝑠
114cv 1521 . . . . . . . 8 class 𝑐
1210, 11wcel 2081 . . . . . . 7 wff 𝑠𝑐
139, 12wi 4 . . . . . 6 wff (𝑠 ≠ ∅ → 𝑠𝑐)
1411cpw 4453 . . . . . 6 class 𝒫 𝑐
1513, 6, 14wral 3105 . . . . 5 wff 𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐)
165, 15wa 396 . . . 4 wff (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))
172cv 1521 . . . . . 6 class 𝑥
1817cpw 4453 . . . . 5 class 𝒫 𝑥
1918cpw 4453 . . . 4 class 𝒫 𝒫 𝑥
2016, 4, 19crab 3109 . . 3 class {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))}
212, 3, 20cmpt 5041 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
221, 21wceq 1522 1 wff Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
Colors of variables: wff setvar class
This definition is referenced by:  ismre  16690  fnmre  16691
  Copyright terms: Public domain W3C validator