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Metamath Proof Explorer

Theorem clsfval

Description: The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis cldval.1 ⊢ 𝑋 = ∪ 𝐽
Assertion clsfval ( 𝐽 ∈ Top → ( cls ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∩ { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥 ⊆ 𝑦 } ) )

Proof

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 } ) )