Definition df-bj-moore

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

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

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

Note: df-mre singles out the empty intersection. This is not necessary. It could be written instead |- Moore = ( x e. _V |-> { y e. ~P ~P x | A. z e. ~P y ( x i^i |^| z ) e. y } ) and the equivalence of both definitions is proved by bj-0int .

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 ).

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 and df-topon ) or the class of all families of that kind, independent of a base set (like df-bj-moore or df-top ). 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)

	Ref	Expression
Assertion	df-bj-moore	|- Moore_ = { x | A. y e. ~P x ( U. x i^i |^| y ) e. x }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cmoore	|- Moore_
1		vx	|- x
2		vy	|- y
3	1	cv	|- x
4	3	cpw	|- ~P x
5	3	cuni	|- U. x
6	2	cv	|- y
7	6	cint	|- |^| y
8	5 7	cin	|- ( U. x i^i |^| y )
9	8 3	wcel	|- ( U. x i^i |^| y ) e. x
10	9 2 4	wral	|- A. y e. ~P x ( U. x i^i |^| y ) e. x
11	10 1	cab	|- { x | A. y e. ~P x ( U. x i^i |^| y ) e. x }
12	0 11	wceq	|- Moore_ = { x | A. y e. ~P x ( U. x i^i |^| y ) e. x }