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 |
|
imastopn.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 ) |
5 |
|
imastopn.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑅 ) |
6 |
|
imastopn.o |
⊢ 𝑂 = ( TopOpen ‘ 𝑈 ) |
7 |
|
eqid |
⊢ ( TopSet ‘ 𝑈 ) = ( TopSet ‘ 𝑈 ) |
8 |
1 2 3 4 5 7
|
imastset |
⊢ ( 𝜑 → ( TopSet ‘ 𝑈 ) = ( 𝐽 qTop 𝐹 ) ) |
9 |
5
|
fvexi |
⊢ 𝐽 ∈ V |
10 |
|
fofn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → 𝐹 Fn 𝑉 ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → 𝐹 Fn 𝑉 ) |
12 |
|
fvex |
⊢ ( Base ‘ 𝑅 ) ∈ V |
13 |
2 12
|
eqeltrdi |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
14 |
|
fnex |
⊢ ( ( 𝐹 Fn 𝑉 ∧ 𝑉 ∈ V ) → 𝐹 ∈ V ) |
15 |
11 13 14
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
16 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
17 |
16
|
qtopval |
⊢ ( ( 𝐽 ∈ V ∧ 𝐹 ∈ V ) → ( 𝐽 qTop 𝐹 ) = { 𝑥 ∈ 𝒫 ( 𝐹 “ ∪ 𝐽 ) ∣ ( ( ◡ 𝐹 “ 𝑥 ) ∩ ∪ 𝐽 ) ∈ 𝐽 } ) |
18 |
9 15 17
|
sylancr |
⊢ ( 𝜑 → ( 𝐽 qTop 𝐹 ) = { 𝑥 ∈ 𝒫 ( 𝐹 “ ∪ 𝐽 ) ∣ ( ( ◡ 𝐹 “ 𝑥 ) ∩ ∪ 𝐽 ) ∈ 𝐽 } ) |
19 |
8 18
|
eqtrd |
⊢ ( 𝜑 → ( TopSet ‘ 𝑈 ) = { 𝑥 ∈ 𝒫 ( 𝐹 “ ∪ 𝐽 ) ∣ ( ( ◡ 𝐹 “ 𝑥 ) ∩ ∪ 𝐽 ) ∈ 𝐽 } ) |
20 |
|
ssrab2 |
⊢ { 𝑥 ∈ 𝒫 ( 𝐹 “ ∪ 𝐽 ) ∣ ( ( ◡ 𝐹 “ 𝑥 ) ∩ ∪ 𝐽 ) ∈ 𝐽 } ⊆ 𝒫 ( 𝐹 “ ∪ 𝐽 ) |
21 |
|
imassrn |
⊢ ( 𝐹 “ ∪ 𝐽 ) ⊆ ran 𝐹 |
22 |
|
forn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
23 |
3 22
|
syl |
⊢ ( 𝜑 → ran 𝐹 = 𝐵 ) |
24 |
1 2 3 4
|
imasbas |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑈 ) ) |
25 |
23 24
|
eqtrd |
⊢ ( 𝜑 → ran 𝐹 = ( Base ‘ 𝑈 ) ) |
26 |
21 25
|
sseqtrid |
⊢ ( 𝜑 → ( 𝐹 “ ∪ 𝐽 ) ⊆ ( Base ‘ 𝑈 ) ) |
27 |
26
|
sspwd |
⊢ ( 𝜑 → 𝒫 ( 𝐹 “ ∪ 𝐽 ) ⊆ 𝒫 ( Base ‘ 𝑈 ) ) |
28 |
20 27
|
sstrid |
⊢ ( 𝜑 → { 𝑥 ∈ 𝒫 ( 𝐹 “ ∪ 𝐽 ) ∣ ( ( ◡ 𝐹 “ 𝑥 ) ∩ ∪ 𝐽 ) ∈ 𝐽 } ⊆ 𝒫 ( Base ‘ 𝑈 ) ) |
29 |
19 28
|
eqsstrd |
⊢ ( 𝜑 → ( TopSet ‘ 𝑈 ) ⊆ 𝒫 ( Base ‘ 𝑈 ) ) |
30 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
31 |
30 7
|
topnid |
⊢ ( ( TopSet ‘ 𝑈 ) ⊆ 𝒫 ( Base ‘ 𝑈 ) → ( TopSet ‘ 𝑈 ) = ( TopOpen ‘ 𝑈 ) ) |
32 |
29 31
|
syl |
⊢ ( 𝜑 → ( TopSet ‘ 𝑈 ) = ( TopOpen ‘ 𝑈 ) ) |
33 |
32 6
|
eqtr4di |
⊢ ( 𝜑 → ( TopSet ‘ 𝑈 ) = 𝑂 ) |
34 |
33 8
|
eqtr3d |
⊢ ( 𝜑 → 𝑂 = ( 𝐽 qTop 𝐹 ) ) |