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Definition df-mopn 19953
Description: Define a function whose value is the family of open sets of a metric space. See elmopn 22464 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 19947 . 2 class MetOpen
2 vd . . 3 setvar 𝑑
3 cxmt 19942 . . . . 5 class ∞Met
43crn 5319 . . . 4 class ran ∞Met
54cuni 4637 . . 3 class ran ∞Met
62cv 1636 . . . . . 6 class 𝑑
7 cbl 19944 . . . . . 6 class ball
86, 7cfv 6104 . . . . 5 class (ball‘𝑑)
98crn 5319 . . . 4 class ran (ball‘𝑑)
10 ctg 16306 . . . 4 class topGen
119, 10cfv 6104 . . 3 class (topGen‘ran (ball‘𝑑))
122, 5, 11cmpt 4930 . 2 class (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
131, 12wceq 1637 1 wff MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Colors of variables: wff setvar class
This definition is referenced by:  mopnval  22460  isxms2  22470  setsmstopn  22500  tngtopn  22671
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