MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mopn Structured version   Visualization version   GIF version

Definition df-mopn 19511
Description: Define a function whose value is the family of open sets of a metric space. See elmopn 22004 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 19505 . 2 class MetOpen
2 vd . . 3 setvar 𝑑
3 cxmt 19500 . . . . 5 class ∞Met
43crn 5028 . . . 4 class ran ∞Met
54cuni 4366 . . 3 class ran ∞Met
62cv 1473 . . . . . 6 class 𝑑
7 cbl 19502 . . . . . 6 class ball
86, 7cfv 5789 . . . . 5 class (ball‘𝑑)
98crn 5028 . . . 4 class ran (ball‘𝑑)
10 ctg 15869 . . . 4 class topGen
119, 10cfv 5789 . . 3 class (topGen‘ran (ball‘𝑑))
122, 5, 11cmpt 4637 . 2 class (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
131, 12wceq 1474 1 wff MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Colors of variables: wff setvar class
This definition is referenced by:  mopnval  22000  isxms2  22010  setsmstopn  22040  tngtopn  22211
  Copyright terms: Public domain W3C validator