Step |
Hyp |
Ref |
Expression |
1 |
|
cldval.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 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) ∈ V ) |
5 |
2 3 4
|
3syl |
⊢ ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) ∈ 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 |
|
fveq2 |
⊢ ( 𝑗 = 𝐽 → ( Clsd ‘ 𝑗 ) = ( Clsd ‘ 𝐽 ) ) |
10 |
9
|
rabeqdv |
⊢ ( 𝑗 = 𝐽 → { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥 ⊆ 𝑦 } = { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) |
11 |
10
|
inteqd |
⊢ ( 𝑗 = 𝐽 → ∩ { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥 ⊆ 𝑦 } = ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) |
12 |
8 11
|
mpteq12dv |
⊢ ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥 ⊆ 𝑦 } ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) ) |
13 |
|
df-cls |
⊢ cls = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥 ⊆ 𝑦 } ) ) |
14 |
12 13
|
fvmptg |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) ∈ V ) → ( cls ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) ) |
15 |
5 14
|
mpdan |
⊢ ( 𝐽 ∈ Top → ( cls ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) ) |