Step |
Hyp |
Ref |
Expression |
1 |
|
mopnval.1 |
β’ π½ = ( MetOpen β π· ) |
2 |
|
fvssunirn |
β’ ( βMet β π ) β βͺ ran βMet |
3 |
2
|
sseli |
β’ ( π· β ( βMet β π ) β π· β βͺ ran βMet ) |
4 |
|
fveq2 |
β’ ( π = π· β ( ball β π ) = ( ball β π· ) ) |
5 |
4
|
rneqd |
β’ ( π = π· β ran ( ball β π ) = ran ( ball β π· ) ) |
6 |
5
|
fveq2d |
β’ ( π = π· β ( topGen β ran ( ball β π ) ) = ( topGen β ran ( ball β π· ) ) ) |
7 |
|
df-mopn |
β’ MetOpen = ( π β βͺ ran βMet β¦ ( topGen β ran ( ball β π ) ) ) |
8 |
|
fvex |
β’ ( topGen β ran ( ball β π· ) ) β V |
9 |
6 7 8
|
fvmpt |
β’ ( π· β βͺ ran βMet β ( MetOpen β π· ) = ( topGen β ran ( ball β π· ) ) ) |
10 |
1 9
|
eqtrid |
β’ ( π· β βͺ ran βMet β π½ = ( topGen β ran ( ball β π· ) ) ) |
11 |
3 10
|
syl |
β’ ( π· β ( βMet β π ) β π½ = ( topGen β ran ( ball β π· ) ) ) |