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Definition df-bj-moore 34514
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 34512, and as illustrated by the lack of sethood condition in bj-ismoore 34515.

This is to df-mre 16852 (defining Moore) what df-top 21502 (defining Top) is to df-topon 21519 (defining TopOn).

For the sake of consistency, the function defined at df-mre 16852 should be denoted by "MooreOn".

Note: df-mre 16852 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 34511.

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 34512).

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 16852 and df-topon 21519) or the class of all families of that kind, independent of a base set (like df-bj-moore 34514 or df-top 21502). 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
StepHypRef Expression
1 cmoore 34513 . 2 class Moore
2 vx . . . . . . . 8 setvar 𝑥
32cv 1537 . . . . . . 7 class 𝑥
43cuni 4803 . . . . . 6 class 𝑥
5 vy . . . . . . . 8 setvar 𝑦
65cv 1537 . . . . . . 7 class 𝑦
76cint 4841 . . . . . 6 class 𝑦
84, 7cin 3883 . . . . 5 class ( 𝑥 𝑦)
98, 3wcel 2112 . . . 4 wff ( 𝑥 𝑦) ∈ 𝑥
103cpw 4500 . . . 4 class 𝒫 𝑥
119, 5, 10wral 3109 . . 3 wff 𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥
1211, 2cab 2779 . 2 class {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥}
131, 12wceq 1538 1 wff Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥}
Colors of variables: wff setvar class
This definition is referenced by:  bj-ismoore  34515
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