Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) |
2 |
1
|
iscmp |
⊢ ( ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Comp ↔ ( ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Top ∧ ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) ) ) |
3 |
2
|
simprbi |
⊢ ( ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Comp → ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) ) |
4 |
|
simpr |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝑋 = ∪ 𝐵 ) |
5 |
|
elex |
⊢ ( 𝑋 ∈ UFL → 𝑋 ∈ V ) |
6 |
5
|
adantr |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝑋 ∈ V ) |
7 |
4 6
|
eqeltrrd |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ∪ 𝐵 ∈ V ) |
8 |
|
uniexb |
⊢ ( 𝐵 ∈ V ↔ ∪ 𝐵 ∈ V ) |
9 |
7 8
|
sylibr |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝐵 ∈ V ) |
10 |
|
fiuni |
⊢ ( 𝐵 ∈ V → ∪ 𝐵 = ∪ ( fi ‘ 𝐵 ) ) |
11 |
9 10
|
syl |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ∪ 𝐵 = ∪ ( fi ‘ 𝐵 ) ) |
12 |
|
fibas |
⊢ ( fi ‘ 𝐵 ) ∈ TopBases |
13 |
|
unitg |
⊢ ( ( fi ‘ 𝐵 ) ∈ TopBases → ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ ( fi ‘ 𝐵 ) ) |
14 |
12 13
|
mp1i |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ ( fi ‘ 𝐵 ) ) |
15 |
11 4 14
|
3eqtr4d |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝑋 = ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) ) |
16 |
15
|
eqeq1d |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( 𝑋 = ∪ 𝑥 ↔ ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑥 ) ) |
17 |
15
|
eqeq1d |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( 𝑋 = ∪ 𝑦 ↔ ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) ) |
18 |
17
|
rexbidv |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ↔ ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) ) |
19 |
16 18
|
imbi12d |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ↔ ( ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) ) ) |
20 |
19
|
ralbidv |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ↔ ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) ) ) |
21 |
|
ssfii |
⊢ ( 𝐵 ∈ V → 𝐵 ⊆ ( fi ‘ 𝐵 ) ) |
22 |
9 21
|
syl |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝐵 ⊆ ( fi ‘ 𝐵 ) ) |
23 |
|
bastg |
⊢ ( ( fi ‘ 𝐵 ) ∈ TopBases → ( fi ‘ 𝐵 ) ⊆ ( topGen ‘ ( fi ‘ 𝐵 ) ) ) |
24 |
12 23
|
ax-mp |
⊢ ( fi ‘ 𝐵 ) ⊆ ( topGen ‘ ( fi ‘ 𝐵 ) ) |
25 |
22 24
|
sstrdi |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝐵 ⊆ ( topGen ‘ ( fi ‘ 𝐵 ) ) ) |
26 |
25
|
sspwd |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → 𝒫 𝐵 ⊆ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ) |
27 |
|
ssralv |
⊢ ( 𝒫 𝐵 ⊆ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) → ( ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) → ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
28 |
26 27
|
syl |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) → ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
29 |
20 28
|
sylbird |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ∀ 𝑥 ∈ 𝒫 ( topGen ‘ ( fi ‘ 𝐵 ) ) ( ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) ∪ ( topGen ‘ ( fi ‘ 𝐵 ) ) = ∪ 𝑦 ) → ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
30 |
3 29
|
syl5 |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Comp → ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
31 |
|
simpll |
⊢ ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) → 𝑋 ∈ UFL ) |
32 |
|
simplr |
⊢ ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) → 𝑋 = ∪ 𝐵 ) |
33 |
|
eqidd |
⊢ ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) → ( topGen ‘ ( fi ‘ 𝐵 ) ) = ( topGen ‘ ( fi ‘ 𝐵 ) ) ) |
34 |
|
velpw |
⊢ ( 𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵 ) |
35 |
|
unieq |
⊢ ( 𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧 ) |
36 |
35
|
eqeq2d |
⊢ ( 𝑥 = 𝑧 → ( 𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ 𝑧 ) ) |
37 |
|
pweq |
⊢ ( 𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧 ) |
38 |
37
|
ineq1d |
⊢ ( 𝑥 = 𝑧 → ( 𝒫 𝑥 ∩ Fin ) = ( 𝒫 𝑧 ∩ Fin ) ) |
39 |
38
|
rexeqdv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ↔ ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) |
40 |
36 39
|
imbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ↔ ( 𝑋 = ∪ 𝑧 → ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
41 |
40
|
rspccv |
⊢ ( ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) → ( 𝑧 ∈ 𝒫 𝐵 → ( 𝑋 = ∪ 𝑧 → ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
42 |
41
|
adantl |
⊢ ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) → ( 𝑧 ∈ 𝒫 𝐵 → ( 𝑋 = ∪ 𝑧 → ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
43 |
34 42
|
syl5bir |
⊢ ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) → ( 𝑧 ⊆ 𝐵 → ( 𝑋 = ∪ 𝑧 → ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
44 |
43
|
imp32 |
⊢ ( ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ∧ ( 𝑧 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑧 ) ) → ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ) |
45 |
|
unieq |
⊢ ( 𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤 ) |
46 |
45
|
eqeq2d |
⊢ ( 𝑦 = 𝑤 → ( 𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑤 ) ) |
47 |
46
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑦 ↔ ∃ 𝑤 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑤 ) |
48 |
44 47
|
sylib |
⊢ ( ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ∧ ( 𝑧 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑧 ) ) → ∃ 𝑤 ∈ ( 𝒫 𝑧 ∩ Fin ) 𝑋 = ∪ 𝑤 ) |
49 |
31 32 33 48
|
alexsub |
⊢ ( ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) → ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Comp ) |
50 |
49
|
ex |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) → ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Comp ) ) |
51 |
30 50
|
impbid |
⊢ ( ( 𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵 ) → ( ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ Comp ↔ ∀ 𝑥 ∈ 𝒫 𝐵 ( 𝑋 = ∪ 𝑥 → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |