Step |
Hyp |
Ref |
Expression |
1 |
|
isfne.1 |
⊢ 𝑋 = ∪ 𝐴 |
2 |
|
isfne.2 |
⊢ 𝑌 = ∪ 𝐵 |
3 |
|
fnerel |
⊢ Rel Fne |
4 |
3
|
brrelex2i |
⊢ ( 𝐴 Fne 𝐵 → 𝐵 ∈ V ) |
5 |
|
simpl |
⊢ ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝑋 = 𝑌 ) |
6 |
5 1 2
|
3eqtr3g |
⊢ ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → ∪ 𝐴 = ∪ 𝐵 ) |
7 |
|
fvex |
⊢ ( topGen ‘ 𝐵 ) ∈ V |
8 |
7
|
ssex |
⊢ ( 𝐴 ⊆ ( topGen ‘ 𝐵 ) → 𝐴 ∈ V ) |
9 |
8
|
adantl |
⊢ ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐴 ∈ V ) |
10 |
9
|
uniexd |
⊢ ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → ∪ 𝐴 ∈ V ) |
11 |
6 10
|
eqeltrrd |
⊢ ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → ∪ 𝐵 ∈ V ) |
12 |
|
uniexb |
⊢ ( 𝐵 ∈ V ↔ ∪ 𝐵 ∈ V ) |
13 |
11 12
|
sylibr |
⊢ ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐵 ∈ V ) |
14 |
1 2
|
isfne |
⊢ ( 𝐵 ∈ V → ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌 ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ) ) ) |
15 |
|
dfss3 |
⊢ ( 𝐴 ⊆ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ( topGen ‘ 𝐵 ) ) |
16 |
|
eltg |
⊢ ( 𝐵 ∈ V → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ) ) |
17 |
16
|
ralbidv |
⊢ ( 𝐵 ∈ V → ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ) ) |
18 |
15 17
|
syl5bb |
⊢ ( 𝐵 ∈ V → ( 𝐴 ⊆ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ) ) |
19 |
18
|
anbi2d |
⊢ ( 𝐵 ∈ V → ( ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) ↔ ( 𝑋 = 𝑌 ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ ∪ ( 𝐵 ∩ 𝒫 𝑥 ) ) ) ) |
20 |
14 19
|
bitr4d |
⊢ ( 𝐵 ∈ V → ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) ) ) |
21 |
4 13 20
|
pm5.21nii |
⊢ ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌 ∧ 𝐴 ⊆ ( topGen ‘ 𝐵 ) ) ) |