Step |
Hyp |
Ref |
Expression |
1 |
|
isnlly |
⊢ ( 𝐽 ∈ 𝑛-Locally 𝐴 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝑥 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑥 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) |
2 |
1
|
simprbi |
⊢ ( 𝐽 ∈ 𝑛-Locally 𝐴 → ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝑥 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑥 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) |
3 |
|
pweq |
⊢ ( 𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈 ) |
4 |
3
|
ineq2d |
⊢ ( 𝑥 = 𝑈 → ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑥 ) = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ) |
5 |
4
|
rexeqdv |
⊢ ( 𝑥 = 𝑈 → ( ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑥 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ↔ ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) |
6 |
5
|
raleqbi1dv |
⊢ ( 𝑥 = 𝑈 → ( ∀ 𝑦 ∈ 𝑥 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑥 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ↔ ∀ 𝑦 ∈ 𝑈 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) |
7 |
6
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝑥 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑥 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ∧ 𝑈 ∈ 𝐽 ) → ∀ 𝑦 ∈ 𝑈 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) |
8 |
2 7
|
sylan |
⊢ ( ( 𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ) → ∀ 𝑦 ∈ 𝑈 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) |
9 |
|
elin |
⊢ ( 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ↔ ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∧ 𝑢 ∈ 𝒫 𝑈 ) ) |
10 |
|
sneq |
⊢ ( 𝑦 = 𝑃 → { 𝑦 } = { 𝑃 } ) |
11 |
10
|
fveq2d |
⊢ ( 𝑦 = 𝑃 → ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) = ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ) |
12 |
11
|
eleq2d |
⊢ ( 𝑦 = 𝑃 → ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ↔ 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ) ) |
13 |
|
velpw |
⊢ ( 𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈 ) |
14 |
13
|
a1i |
⊢ ( 𝑦 = 𝑃 → ( 𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈 ) ) |
15 |
12 14
|
anbi12d |
⊢ ( 𝑦 = 𝑃 → ( ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∧ 𝑢 ∈ 𝒫 𝑈 ) ↔ ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∧ 𝑢 ⊆ 𝑈 ) ) ) |
16 |
9 15
|
syl5bb |
⊢ ( 𝑦 = 𝑃 → ( 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ↔ ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∧ 𝑢 ⊆ 𝑈 ) ) ) |
17 |
16
|
anbi1d |
⊢ ( 𝑦 = 𝑃 → ( ( 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ↔ ( ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∧ 𝑢 ⊆ 𝑈 ) ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) ) |
18 |
|
anass |
⊢ ( ( ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∧ 𝑢 ⊆ 𝑈 ) ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ↔ ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∧ ( 𝑢 ⊆ 𝑈 ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) ) |
19 |
17 18
|
bitrdi |
⊢ ( 𝑦 = 𝑃 → ( ( 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ↔ ( 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∧ ( 𝑢 ⊆ 𝑈 ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) ) ) |
20 |
19
|
rexbidv2 |
⊢ ( 𝑦 = 𝑃 → ( ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ↔ ∃ 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑢 ⊆ 𝑈 ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) ) |
21 |
20
|
rspccva |
⊢ ( ( ∀ 𝑦 ∈ 𝑈 ∃ 𝑢 ∈ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑦 } ) ∩ 𝒫 𝑈 ) ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ∧ 𝑃 ∈ 𝑈 ) → ∃ 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑢 ⊆ 𝑈 ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) |
22 |
8 21
|
stoic3 |
⊢ ( ( 𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈 ) → ∃ 𝑢 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑢 ⊆ 𝑈 ∧ ( 𝐽 ↾t 𝑢 ) ∈ 𝐴 ) ) |