Step |
Hyp |
Ref |
Expression |
1 |
|
tgval |
⊢ ( 𝐵 ∈ 𝑉 → ( topGen ‘ 𝐵 ) = { 𝑥 ∣ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) } ) |
2 |
1
|
eleq2d |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ 𝐴 ∈ { 𝑥 ∣ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) } ) ) |
3 |
|
elex |
⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) } → 𝐴 ∈ V ) |
4 |
3
|
adantl |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ { 𝑥 ∣ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) } ) → 𝐴 ∈ V ) |
5 |
|
inex1g |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∩ 𝒫 𝐴 ) ∈ V ) |
6 |
5
|
uniexd |
⊢ ( 𝐵 ∈ 𝑉 → ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ∈ V ) |
7 |
|
ssexg |
⊢ ( ( 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ∧ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ∈ V ) → 𝐴 ∈ V ) |
8 |
6 7
|
sylan2 |
⊢ ( ( 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ V ) |
9 |
8
|
ancoms |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ) → 𝐴 ∈ V ) |
10 |
|
id |
⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 ) |
11 |
|
pweq |
⊢ ( 𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴 ) |
12 |
11
|
ineq2d |
⊢ ( 𝑥 = 𝐴 → ( 𝐵 ∩ 𝒫 𝑥 ) = ( 𝐵 ∩ 𝒫 𝐴 ) ) |
13 |
12
|
unieqd |
⊢ ( 𝑥 = 𝐴 → ∪ ( 𝐵 ∩ 𝒫 𝑥 ) = ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ) |
14 |
10 13
|
sseq12d |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ↔ 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ) ) |
15 |
14
|
elabg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝑥 ∣ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) } ↔ 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ) ) |
16 |
4 9 15
|
pm5.21nd |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ { 𝑥 ∣ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) } ↔ 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ) ) |
17 |
2 16
|
bitrd |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ 𝐴 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝐴 ) ) ) |