Definition df-mopn 21360

Description: Define a function whose value is the family of open sets of a metric space. See elmopn 24452 for its main property. (Contributed by NM, 1-Sep-2006.)

Assertion

Ref	Expression
df-mopn	⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))

Detailed syntax breakdown of Definition df-mopn

Step	Hyp	Ref	Expression
1		cmopn 21354	. 2 class MetOpen
2		vd	. . 3 setvar 𝑑
3		cxmet 21349	. . . . 5 class ∞Met
4	3	crn 5686	. . . 4 class ran ∞Met
5	4	cuni 4907	. . 3 class ∪ ran ∞Met
6	2	cv 1539	. . . . . 6 class 𝑑
7		cbl 21351	. . . . . 6 class ball
8	6, 7	cfv 6561	. . . . 5 class (ball‘𝑑)
9	8	crn 5686	. . . 4 class ran (ball‘𝑑)
10		ctg 17482	. . . 4 class topGen
11	9, 10	cfv 6561	. . 3 class (topGen‘ran (ball‘𝑑))
12	2, 5, 11	cmpt 5225	. 2 class (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
13	1, 12	wceq 1540	1 wff MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))

This definition is referenced by:  mopnval  24448  isxms2  24458  setsmstopn  24490  tngtopn  24671