Step |
Hyp |
Ref |
Expression |
1 |
|
topfneec.1 |
⊢ ∼ = ( Fne ∩ ◡ Fne ) |
2 |
1
|
fneer |
⊢ ∼ Er V |
3 |
|
errel |
⊢ ( ∼ Er V → Rel ∼ ) |
4 |
2 3
|
ax-mp |
⊢ Rel ∼ |
5 |
|
relelec |
⊢ ( Rel ∼ → ( 𝐴 ∈ [ 𝐽 ] ∼ ↔ 𝐽 ∼ 𝐴 ) ) |
6 |
4 5
|
ax-mp |
⊢ ( 𝐴 ∈ [ 𝐽 ] ∼ ↔ 𝐽 ∼ 𝐴 ) |
7 |
4
|
brrelex2i |
⊢ ( 𝐽 ∼ 𝐴 → 𝐴 ∈ V ) |
8 |
7
|
a1i |
⊢ ( 𝐽 ∈ Top → ( 𝐽 ∼ 𝐴 → 𝐴 ∈ V ) ) |
9 |
|
eleq1 |
⊢ ( ( topGen ‘ 𝐴 ) = 𝐽 → ( ( topGen ‘ 𝐴 ) ∈ Top ↔ 𝐽 ∈ Top ) ) |
10 |
9
|
biimparc |
⊢ ( ( 𝐽 ∈ Top ∧ ( topGen ‘ 𝐴 ) = 𝐽 ) → ( topGen ‘ 𝐴 ) ∈ Top ) |
11 |
|
tgclb |
⊢ ( 𝐴 ∈ TopBases ↔ ( topGen ‘ 𝐴 ) ∈ Top ) |
12 |
10 11
|
sylibr |
⊢ ( ( 𝐽 ∈ Top ∧ ( topGen ‘ 𝐴 ) = 𝐽 ) → 𝐴 ∈ TopBases ) |
13 |
|
elex |
⊢ ( 𝐴 ∈ TopBases → 𝐴 ∈ V ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐽 ∈ Top ∧ ( topGen ‘ 𝐴 ) = 𝐽 ) → 𝐴 ∈ V ) |
15 |
14
|
ex |
⊢ ( 𝐽 ∈ Top → ( ( topGen ‘ 𝐴 ) = 𝐽 → 𝐴 ∈ V ) ) |
16 |
1
|
fneval |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) → ( 𝐽 ∼ 𝐴 ↔ ( topGen ‘ 𝐽 ) = ( topGen ‘ 𝐴 ) ) ) |
17 |
|
tgtop |
⊢ ( 𝐽 ∈ Top → ( topGen ‘ 𝐽 ) = 𝐽 ) |
18 |
17
|
eqeq1d |
⊢ ( 𝐽 ∈ Top → ( ( topGen ‘ 𝐽 ) = ( topGen ‘ 𝐴 ) ↔ 𝐽 = ( topGen ‘ 𝐴 ) ) ) |
19 |
|
eqcom |
⊢ ( 𝐽 = ( topGen ‘ 𝐴 ) ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) |
20 |
18 19
|
syl6bb |
⊢ ( 𝐽 ∈ Top → ( ( topGen ‘ 𝐽 ) = ( topGen ‘ 𝐴 ) ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) ) |
21 |
20
|
adantr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) → ( ( topGen ‘ 𝐽 ) = ( topGen ‘ 𝐴 ) ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) ) |
22 |
16 21
|
bitrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) → ( 𝐽 ∼ 𝐴 ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) ) |
23 |
22
|
ex |
⊢ ( 𝐽 ∈ Top → ( 𝐴 ∈ V → ( 𝐽 ∼ 𝐴 ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) ) ) |
24 |
8 15 23
|
pm5.21ndd |
⊢ ( 𝐽 ∈ Top → ( 𝐽 ∼ 𝐴 ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) ) |
25 |
6 24
|
syl5bb |
⊢ ( 𝐽 ∈ Top → ( 𝐴 ∈ [ 𝐽 ] ∼ ↔ ( topGen ‘ 𝐴 ) = 𝐽 ) ) |