Step |
Hyp |
Ref |
Expression |
1 |
|
tpspropd.1 |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) ) |
2 |
|
tpspropd.2 |
⊢ ( 𝜑 → ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐿 ) ) |
3 |
1
|
fveq2d |
⊢ ( 𝜑 → ( TopOn ‘ ( Base ‘ 𝐾 ) ) = ( TopOn ‘ ( Base ‘ 𝐿 ) ) ) |
4 |
2 3
|
eleq12d |
⊢ ( 𝜑 → ( ( TopOpen ‘ 𝐾 ) ∈ ( TopOn ‘ ( Base ‘ 𝐾 ) ) ↔ ( TopOpen ‘ 𝐿 ) ∈ ( TopOn ‘ ( Base ‘ 𝐿 ) ) ) ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 ) |
7 |
5 6
|
istps |
⊢ ( 𝐾 ∈ TopSp ↔ ( TopOpen ‘ 𝐾 ) ∈ ( TopOn ‘ ( Base ‘ 𝐾 ) ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
9 |
|
eqid |
⊢ ( TopOpen ‘ 𝐿 ) = ( TopOpen ‘ 𝐿 ) |
10 |
8 9
|
istps |
⊢ ( 𝐿 ∈ TopSp ↔ ( TopOpen ‘ 𝐿 ) ∈ ( TopOn ‘ ( Base ‘ 𝐿 ) ) ) |
11 |
4 7 10
|
3bitr4g |
⊢ ( 𝜑 → ( 𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp ) ) |