Step |
Hyp |
Ref |
Expression |
1 |
|
imastps.u |
⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) ) |
2 |
|
imastps.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) ) |
3 |
|
imastps.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
4 |
|
imastps.r |
⊢ ( 𝜑 → 𝑅 ∈ TopSp ) |
5 |
|
eqid |
⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( TopOpen ‘ 𝑈 ) = ( TopOpen ‘ 𝑈 ) |
7 |
1 2 3 4 5 6
|
imastopn |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑈 ) = ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
8 5
|
istps |
⊢ ( 𝑅 ∈ TopSp ↔ ( TopOpen ‘ 𝑅 ) ∈ ( TopOn ‘ ( Base ‘ 𝑅 ) ) ) |
10 |
4 9
|
sylib |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑅 ) ∈ ( TopOn ‘ ( Base ‘ 𝑅 ) ) ) |
11 |
2
|
fveq2d |
⊢ ( 𝜑 → ( TopOn ‘ 𝑉 ) = ( TopOn ‘ ( Base ‘ 𝑅 ) ) ) |
12 |
10 11
|
eleqtrrd |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑅 ) ∈ ( TopOn ‘ 𝑉 ) ) |
13 |
|
qtoptopon |
⊢ ( ( ( TopOpen ‘ 𝑅 ) ∈ ( TopOn ‘ 𝑉 ) ∧ 𝐹 : 𝑉 –onto→ 𝐵 ) → ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ∈ ( TopOn ‘ 𝐵 ) ) |
14 |
12 3 13
|
syl2anc |
⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ∈ ( TopOn ‘ 𝐵 ) ) |
15 |
1 2 3 4
|
imasbas |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑈 ) ) |
16 |
15
|
fveq2d |
⊢ ( 𝜑 → ( TopOn ‘ 𝐵 ) = ( TopOn ‘ ( Base ‘ 𝑈 ) ) ) |
17 |
14 16
|
eleqtrd |
⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ∈ ( TopOn ‘ ( Base ‘ 𝑈 ) ) ) |
18 |
7 17
|
eqeltrd |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑈 ) ∈ ( TopOn ‘ ( Base ‘ 𝑈 ) ) ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
20 |
19 6
|
istps |
⊢ ( 𝑈 ∈ TopSp ↔ ( TopOpen ‘ 𝑈 ) ∈ ( TopOn ‘ ( Base ‘ 𝑈 ) ) ) |
21 |
18 20
|
sylibr |
⊢ ( 𝜑 → 𝑈 ∈ TopSp ) |