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Mirrors > Home > MPE Home > Th. List > df-mopn | Structured version Visualization version GIF version |
Description: Define a function whose value is the family of open sets of a metric space. See elmopn 23604 for its main property. (Contributed by NM, 1-Sep-2006.) |
Ref | Expression |
---|---|
df-mopn | ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmopn 20596 | . 2 class MetOpen | |
2 | vd | . . 3 setvar 𝑑 | |
3 | cxmet 20591 | . . . . 5 class ∞Met | |
4 | 3 | crn 5591 | . . . 4 class ran ∞Met |
5 | 4 | cuni 4840 | . . 3 class ∪ ran ∞Met |
6 | 2 | cv 1538 | . . . . . 6 class 𝑑 |
7 | cbl 20593 | . . . . . 6 class ball | |
8 | 6, 7 | cfv 6437 | . . . . 5 class (ball‘𝑑) |
9 | 8 | crn 5591 | . . . 4 class ran (ball‘𝑑) |
10 | ctg 17157 | . . . 4 class topGen | |
11 | 9, 10 | cfv 6437 | . . 3 class (topGen‘ran (ball‘𝑑)) |
12 | 2, 5, 11 | cmpt 5158 | . 2 class (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) |
13 | 1, 12 | wceq 1539 | 1 wff MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) |
Colors of variables: wff setvar class |
This definition is referenced by: mopnval 23600 isxms2 23610 setsmstopn 23642 tngtopn 23823 |
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