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Definition df-mopn 19563
 Description: Define a function whose value is the family of open sets of a metric space. See elmopn 22057 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 19557 . 2 class MetOpen
2 vd . . 3 setvar 𝑑
3 cxmt 19552 . . . . 5 class ∞Met
43crn 5039 . . . 4 class ran ∞Met
54cuni 4372 . . 3 class ran ∞Met
62cv 1474 . . . . . 6 class 𝑑
7 cbl 19554 . . . . . 6 class ball
86, 7cfv 5804 . . . . 5 class (ball‘𝑑)
98crn 5039 . . . 4 class ran (ball‘𝑑)
10 ctg 15921 . . . 4 class topGen
119, 10cfv 5804 . . 3 class (topGen‘ran (ball‘𝑑))
122, 5, 11cmpt 4643 . 2 class (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
131, 12wceq 1475 1 wff MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
 Colors of variables: wff setvar class This definition is referenced by:  mopnval  22053  isxms2  22063  setsmstopn  22093  tngtopn  22264
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