Definition df-bj-moore 37087

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 37085, and as illustrated by the lack of sethood condition in bj-ismoore 37088.

This is to df-mre 17631 (defining Moore) what df-top 22916 (defining Top) is to df-topon 22933 (defining TopOn).

For the sake of consistency, the function defined at df-mre 17631 should be denoted by "MooreOn".

Note: df-mre 17631 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 37084.

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 37085).

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 17631 and df-topon 22933) or the class of all families of that kind, independent of a base set (like df-bj-moore 37087 or df-top 22916). 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 37086	. 2 class Moore
2		vx	. . . . . . . 8 setvar 𝑥
3	2	cv 1536	. . . . . . 7 class 𝑥
4	3	cuni 4912	. . . . . 6 class ∪ 𝑥
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1536	. . . . . . 7 class 𝑦
7	6	cint 4951	. . . . . 6 class ∩ 𝑦
8	4, 7	cin 3962	. . . . 5 class (∪ 𝑥 ∩ ∩ 𝑦)
9	8, 3	wcel 2106	. . . 4 wff (∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
10	3	cpw 4605	. . . 4 class 𝒫 𝑥
11	9, 5, 10	wral 3059	. . 3 wff ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
12	11, 2	cab 2712	. 2 class {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
13	1, 12	wceq 1537	1 wff Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}

This definition is referenced by:  bj-ismoore  37088