Description: The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | mopnval.1 | |- J = ( MetOpen ` D ) |
|
| Assertion | mopnuni | |- ( D e. ( *Met ` X ) -> X = U. J ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mopnval.1 | |- J = ( MetOpen ` D ) |
|
| 2 | 1 | mopntopon | |- ( D e. ( *Met ` X ) -> J e. ( TopOn ` X ) ) |
| 3 | toponuni | |- ( J e. ( TopOn ` X ) -> X = U. J ) |
|
| 4 | 2 3 | syl | |- ( D e. ( *Met ` X ) -> X = U. J ) |