Step |
Hyp |
Ref |
Expression |
1 |
|
neifval.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
topopn |
⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 ) |
3 |
|
pwexg |
⊢ ( 𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V ) |
4 |
|
mptexg |
⊢ ( 𝒫 𝑋 ∈ V → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ∈ V ) |
5 |
2 3 4
|
3syl |
⊢ ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ∈ V ) |
6 |
|
unieq |
⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 ) |
7 |
6 1
|
eqtr4di |
⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = 𝑋 ) |
8 |
7
|
pweqd |
⊢ ( 𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋 ) |
9 |
|
rexeq |
⊢ ( 𝑗 = 𝐽 → ( ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ↔ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ) ) |
10 |
8 9
|
rabeqbidv |
⊢ ( 𝑗 = 𝐽 → { 𝑣 ∈ 𝒫 ∪ 𝑗 ∣ ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) |
11 |
8 10
|
mpteq12dv |
⊢ ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ { 𝑣 ∈ 𝒫 ∪ 𝑗 ∣ ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ) |
12 |
|
df-nei |
⊢ nei = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ { 𝑣 ∈ 𝒫 ∪ 𝑗 ∣ ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ) |
13 |
11 12
|
fvmptg |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ∈ V ) → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ) |
14 |
5 13
|
mpdan |
⊢ ( 𝐽 ∈ Top → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ) |