Step |
Hyp |
Ref |
Expression |
1 |
|
tpspropd.1 |
|- ( ph -> ( Base ` K ) = ( Base ` L ) ) |
2 |
|
tpspropd.2 |
|- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) ) |
3 |
1
|
fveq2d |
|- ( ph -> ( TopOn ` ( Base ` K ) ) = ( TopOn ` ( Base ` L ) ) ) |
4 |
2 3
|
eleq12d |
|- ( ph -> ( ( TopOpen ` K ) e. ( TopOn ` ( Base ` K ) ) <-> ( TopOpen ` L ) e. ( TopOn ` ( Base ` L ) ) ) ) |
5 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
6 |
|
eqid |
|- ( TopOpen ` K ) = ( TopOpen ` K ) |
7 |
5 6
|
istps |
|- ( K e. TopSp <-> ( TopOpen ` K ) e. ( TopOn ` ( Base ` K ) ) ) |
8 |
|
eqid |
|- ( Base ` L ) = ( Base ` L ) |
9 |
|
eqid |
|- ( TopOpen ` L ) = ( TopOpen ` L ) |
10 |
8 9
|
istps |
|- ( L e. TopSp <-> ( TopOpen ` L ) e. ( TopOn ` ( Base ` L ) ) ) |
11 |
4 7 10
|
3bitr4g |
|- ( ph -> ( K e. TopSp <-> L e. TopSp ) ) |