Step |
Hyp |
Ref |
Expression |
1 |
|
iscvm.1 |
⊢ 𝑆 = ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } ) |
2 |
|
unieq |
⊢ ( 𝑠 = 𝑏 → ∪ 𝑠 = ∪ 𝑏 ) |
3 |
2
|
eqeq1d |
⊢ ( 𝑠 = 𝑏 → ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ↔ ∪ 𝑏 = ( ◡ 𝐹 “ 𝑘 ) ) ) |
4 |
|
ineq2 |
⊢ ( 𝑣 = 𝑑 → ( 𝑢 ∩ 𝑣 ) = ( 𝑢 ∩ 𝑑 ) ) |
5 |
4
|
eqeq1d |
⊢ ( 𝑣 = 𝑑 → ( ( 𝑢 ∩ 𝑣 ) = ∅ ↔ ( 𝑢 ∩ 𝑑 ) = ∅ ) ) |
6 |
5
|
cbvralvw |
⊢ ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ↔ ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑑 ) = ∅ ) |
7 |
|
sneq |
⊢ ( 𝑢 = 𝑐 → { 𝑢 } = { 𝑐 } ) |
8 |
7
|
difeq2d |
⊢ ( 𝑢 = 𝑐 → ( 𝑠 ∖ { 𝑢 } ) = ( 𝑠 ∖ { 𝑐 } ) ) |
9 |
|
ineq1 |
⊢ ( 𝑢 = 𝑐 → ( 𝑢 ∩ 𝑑 ) = ( 𝑐 ∩ 𝑑 ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑢 = 𝑐 → ( ( 𝑢 ∩ 𝑑 ) = ∅ ↔ ( 𝑐 ∩ 𝑑 ) = ∅ ) ) |
11 |
8 10
|
raleqbidv |
⊢ ( 𝑢 = 𝑐 → ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑑 ) = ∅ ↔ ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ) ) |
12 |
6 11
|
syl5bb |
⊢ ( 𝑢 = 𝑐 → ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ↔ ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ) ) |
13 |
|
reseq2 |
⊢ ( 𝑢 = 𝑐 → ( 𝐹 ↾ 𝑢 ) = ( 𝐹 ↾ 𝑐 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑢 = 𝑐 → ( 𝐶 ↾t 𝑢 ) = ( 𝐶 ↾t 𝑐 ) ) |
15 |
14
|
oveq1d |
⊢ ( 𝑢 = 𝑐 → ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) = ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) |
16 |
13 15
|
eleq12d |
⊢ ( 𝑢 = 𝑐 → ( ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ↔ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) |
17 |
12 16
|
anbi12d |
⊢ ( 𝑢 = 𝑐 → ( ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ) |
18 |
17
|
cbvralvw |
⊢ ( ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ∀ 𝑐 ∈ 𝑠 ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) |
19 |
|
difeq1 |
⊢ ( 𝑠 = 𝑏 → ( 𝑠 ∖ { 𝑐 } ) = ( 𝑏 ∖ { 𝑐 } ) ) |
20 |
19
|
raleqdv |
⊢ ( 𝑠 = 𝑏 → ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ↔ ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ) ) |
21 |
20
|
anbi1d |
⊢ ( 𝑠 = 𝑏 → ( ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ) |
22 |
21
|
raleqbi1dv |
⊢ ( 𝑠 = 𝑏 → ( ∀ 𝑐 ∈ 𝑠 ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ) |
23 |
18 22
|
syl5bb |
⊢ ( 𝑠 = 𝑏 → ( ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ) |
24 |
3 23
|
anbi12d |
⊢ ( 𝑠 = 𝑏 → ( ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ↔ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ) ) |
25 |
24
|
cbvrabv |
⊢ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } = { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } |
26 |
|
imaeq2 |
⊢ ( 𝑘 = 𝑎 → ( ◡ 𝐹 “ 𝑘 ) = ( ◡ 𝐹 “ 𝑎 ) ) |
27 |
26
|
eqeq2d |
⊢ ( 𝑘 = 𝑎 → ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑘 ) ↔ ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ) ) |
28 |
|
oveq2 |
⊢ ( 𝑘 = 𝑎 → ( 𝐽 ↾t 𝑘 ) = ( 𝐽 ↾t 𝑎 ) ) |
29 |
28
|
oveq2d |
⊢ ( 𝑘 = 𝑎 → ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) = ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) |
30 |
29
|
eleq2d |
⊢ ( 𝑘 = 𝑎 → ( ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ↔ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) |
31 |
30
|
anbi2d |
⊢ ( 𝑘 = 𝑎 → ( ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) ) |
32 |
31
|
ralbidv |
⊢ ( 𝑘 = 𝑎 → ( ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ↔ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) ) |
33 |
27 32
|
anbi12d |
⊢ ( 𝑘 = 𝑎 → ( ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) ↔ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) ) ) |
34 |
33
|
rabbidv |
⊢ ( 𝑘 = 𝑎 → { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } = { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) } ) |
35 |
25 34
|
syl5eq |
⊢ ( 𝑘 = 𝑎 → { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } = { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) } ) |
36 |
35
|
cbvmptv |
⊢ ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } ) = ( 𝑎 ∈ 𝐽 ↦ { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) } ) |
37 |
1 36
|
eqtri |
⊢ 𝑆 = ( 𝑎 ∈ 𝐽 ↦ { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ∧ ∀ 𝑐 ∈ 𝑏 ( ∀ 𝑑 ∈ ( 𝑏 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) } ) |