Definition df-mopn 21231

Description: Define a function whose value is the family of open sets of a metric space. See elmopn 24298 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 21225	. 2 class MetOpen
2		vd	. . 3 setvar 𝑑
3		cxmet 21220	. . . . 5 class ∞Met
4	3	crn 5670	. . . 4 class ran ∞Met
5	4	cuni 4902	. . 3 class ∪ ran ∞Met
6	2	cv 1532	. . . . . 6 class 𝑑
7		cbl 21222	. . . . . 6 class ball
8	6, 7	cfv 6536	. . . . 5 class (ball‘𝑑)
9	8	crn 5670	. . . 4 class ran (ball‘𝑑)
10		ctg 17389	. . . 4 class topGen
11	9, 10	cfv 6536	. . 3 class (topGen‘ran (ball‘𝑑))
12	2, 5, 11	cmpt 5224	. 2 class (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
13	1, 12	wceq 1533	1 wff MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))

This definition is referenced by:  mopnval  24294  isxms2  24304  setsmstopn  24336  tngtopn  24517