Definition df-bj-moore 37384

Description: Define the class of Moore collections. This is indeed the class of all Moore collections since these all are sets, as proved in bj-mooreset 37382, and as illustrated by the lack of sethood condition in bj-ismoore 37385.

This is to df-mre 17519 (defining Moore) what df-top 22855 (defining Top) is to df-topon 22872 (defining TopOn).

For the sake of consistency, the function defined at df-mre 17519 should be denoted by "MooreOn".

Note: df-mre 17519 singles out the empty intersection. This is not necessary. It could be written instead ⊢ Moore = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝒫 𝑥 ∣ ∀𝑧 ∈ 𝒫 𝑦(𝑥 ∩ ∩ 𝑧) ∈ 𝑦}) and the equivalence of both definitions is proved by bj-0int 37381.

There is no added generality in defining a "Moore predicate" for arbitrary classes, since a Moore class satisfying such a predicate is automatically a set (see bj-mooreset 37382).

TODO: move to the main section. For many families of sets, one can define both the function associating to each set the set of families of that kind on it (like df-mre 17519 and df-topon 22872) or the class of all families of that kind, independent of a base set (like df-bj-moore 37384 or df-top 22855). In general, the former will be more useful and the extra generality of the latter is not necessary. Moore collections, however, are particular in that they are more ubiquitous and are used in a wide variety of applications (for many families of sets, the family of families of a given kind is often a Moore collection, for instance). Therefore, in the case of Moore families, having both definitions is useful.

(Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
df-bj-moore	⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-bj-moore

Step	Hyp	Ref	Expression
1		cmoore 37383	. 2 class Moore
2		vx	. . . . . . . 8 setvar 𝑥
3	2	cv 1541	. . . . . . 7 class 𝑥
4	3	cuni 4865	. . . . . 6 class ∪ 𝑥
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1541	. . . . . . 7 class 𝑦
7	6	cint 4904	. . . . . 6 class ∩ 𝑦
8	4, 7	cin 3902	. . . . 5 class (∪ 𝑥 ∩ ∩ 𝑦)
9	8, 3	wcel 2114	. . . 4 wff (∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
10	3	cpw 4556	. . . 4 class 𝒫 𝑥
11	9, 5, 10	wral 3052	. . 3 wff ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
12	11, 2	cab 2715	. 2 class {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
13	1, 12	wceq 1542	1 wff Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}

This definition is referenced by:  bj-ismoore  37385