Definition df-bj-moore 37404

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 37402, and as illustrated by the lack of sethood condition in bj-ismoore 37405.

This is to df-mre 17537 (defining Moore) what df-top 22847 (defining Top) is to df-topon 22864 (defining TopOn).

For the sake of consistency, the function defined at df-mre 17537 should be denoted by "MooreOn".

Note: df-mre 17537 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 37401.

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 37402).

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 17537 and df-topon 22864) or the class of all families of that kind, independent of a base set (like df-bj-moore 37404 or df-top 22847). 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 37403	. 2 class Moore
2		vx	. . . . . . . 8 setvar 𝑥
3	2	cv 1541	. . . . . . 7 class 𝑥
4	3	cuni 4840	. . . . . 6 class ∪ 𝑥
5		vy	. . . . . . . 8 setvar 𝑦
6	5	cv 1541	. . . . . . 7 class 𝑦
7	6	cint 4879	. . . . . 6 class ∩ 𝑦
8	4, 7	cin 3884	. . . . 5 class (∪ 𝑥 ∩ ∩ 𝑦)
9	8, 3	wcel 2114	. . . 4 wff (∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
10	3	cpw 4531	. . . 4 class 𝒫 𝑥
11	9, 5, 10	wral 3049	. . 3 wff ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥
12	11, 2	cab 2713	. 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  37405