Definition df-bj-moore 37095

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 37093, and as illustrated by the lack of sethood condition in bj-ismoore 37096.

This is to df-mre 17475 (defining Moore) what df-top 22763 (defining Top) is to df-topon 22780 (defining TopOn).

For the sake of consistency, the function defined at df-mre 17475 should be denoted by "MooreOn".

Note: df-mre 17475 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 37092.

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 37093).

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 17475 and df-topon 22780) or the class of all families of that kind, independent of a base set (like df-bj-moore 37095 or df-top 22763). 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 37094	. 2 class Moore
2		vx	. . . . . . . 8 setvar 𝑥
3	2	cv 1539	. . . . . . 7 class 𝑥
4	3	cuni 4856	. . . . . 6 class ∪ 𝑥
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1539	. . . . . . 7 class 𝑦
7	6	cint 4894	. . . . . 6 class ∩ 𝑦
8	4, 7	cin 3898	. . . . 5 class (∪ 𝑥 ∩ ∩ 𝑦)
9	8, 3	wcel 2109	. . . 4 wff (∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
10	3	cpw 4547	. . . 4 class 𝒫 𝑥
11	9, 5, 10	wral 3044	. . 3 wff ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
12	11, 2	cab 2707	. 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  37096