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Definition df-mopn 20091
Description: Define a function whose value is the family of open sets of a metric space. See elmopn 23053 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
StepHypRef Expression
1 cmopn 20085 . 2 class MetOpen
2 vd . . 3 setvar 𝑑
3 cxmet 20080 . . . . 5 class ∞Met
43crn 5524 . . . 4 class ran ∞Met
54cuni 4803 . . 3 class ran ∞Met
62cv 1537 . . . . . 6 class 𝑑
7 cbl 20082 . . . . . 6 class ball
86, 7cfv 6328 . . . . 5 class (ball‘𝑑)
98crn 5524 . . . 4 class ran (ball‘𝑑)
10 ctg 16707 . . . 4 class topGen
119, 10cfv 6328 . . 3 class (topGen‘ran (ball‘𝑑))
122, 5, 11cmpt 5113 . 2 class (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
131, 12wceq 1538 1 wff MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Colors of variables: wff setvar class
This definition is referenced by:  mopnval  23049  isxms2  23059  setsmstopn  23089  tngtopn  23260
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