Step |
Hyp |
Ref |
Expression |
1 |
|
qtoptop.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
ssidd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝑌 ⊆ 𝑌 ) |
3 |
|
fof |
⊢ ( 𝐹 : 𝑋 –onto→ 𝑌 → 𝐹 : 𝑋 ⟶ 𝑌 ) |
4 |
3
|
adantl |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
5 |
|
fimacnv |
⊢ ( 𝐹 : 𝑋 ⟶ 𝑌 → ( ◡ 𝐹 “ 𝑌 ) = 𝑋 ) |
6 |
4 5
|
syl |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( ◡ 𝐹 “ 𝑌 ) = 𝑋 ) |
7 |
1
|
topopn |
⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 ) |
8 |
7
|
adantr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝑋 ∈ 𝐽 ) |
9 |
6 8
|
eqeltrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( ◡ 𝐹 “ 𝑌 ) ∈ 𝐽 ) |
10 |
1
|
elqtop2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝑌 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝑌 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑌 ) ∈ 𝐽 ) ) ) |
11 |
2 9 10
|
mpbir2and |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝑌 ∈ ( 𝐽 qTop 𝐹 ) ) |
12 |
|
elssuni |
⊢ ( 𝑌 ∈ ( 𝐽 qTop 𝐹 ) → 𝑌 ⊆ ∪ ( 𝐽 qTop 𝐹 ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝑌 ⊆ ∪ ( 𝐽 qTop 𝐹 ) ) |
14 |
1
|
elqtop2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝑥 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝑥 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
15 |
|
simpl |
⊢ ( ( 𝑥 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) → 𝑥 ⊆ 𝑌 ) |
16 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌 ) |
17 |
15 16
|
sylibr |
⊢ ( ( 𝑥 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) → 𝑥 ∈ 𝒫 𝑌 ) |
18 |
14 17
|
syl6bi |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝑥 ∈ ( 𝐽 qTop 𝐹 ) → 𝑥 ∈ 𝒫 𝑌 ) ) |
19 |
18
|
ssrdv |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝐽 qTop 𝐹 ) ⊆ 𝒫 𝑌 ) |
20 |
|
sspwuni |
⊢ ( ( 𝐽 qTop 𝐹 ) ⊆ 𝒫 𝑌 ↔ ∪ ( 𝐽 qTop 𝐹 ) ⊆ 𝑌 ) |
21 |
19 20
|
sylib |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ∪ ( 𝐽 qTop 𝐹 ) ⊆ 𝑌 ) |
22 |
13 21
|
eqssd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝑌 = ∪ ( 𝐽 qTop 𝐹 ) ) |