Definition df-bj-moore 37099

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 37097, and as illustrated by the lack of sethood condition in bj-ismoore 37100.

This is to df-mre 17554 (defining Moore) what df-top 22788 (defining Top) is to df-topon 22805 (defining TopOn).

For the sake of consistency, the function defined at df-mre 17554 should be denoted by "MooreOn".

Note: df-mre 17554 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 37096.

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 37097).

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 17554 and df-topon 22805) or the class of all families of that kind, independent of a base set (like df-bj-moore 37099 or df-top 22788). 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 37098	. 2 class Moore
2		vx	. . . . . . . 8 setvar 𝑥
3	2	cv 1539	. . . . . . 7 class 𝑥
4	3	cuni 4874	. . . . . 6 class ∪ 𝑥
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1539	. . . . . . 7 class 𝑦
7	6	cint 4913	. . . . . 6 class ∩ 𝑦
8	4, 7	cin 3916	. . . . 5 class (∪ 𝑥 ∩ ∩ 𝑦)
9	8, 3	wcel 2109	. . . 4 wff (∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
10	3	cpw 4566	. . . 4 class 𝒫 𝑥
11	9, 5, 10	wral 3045	. . 3 wff ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
12	11, 2	cab 2708	. 2 class {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
13	1, 12	wceq 1540	1 wff Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}

This definition is referenced by:  bj-ismoore  37100