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 ) ∣ 𝑥𝑦 } ) )

Proof

Step Hyp Ref Expression
1 cldval.1 𝑋 = 𝐽
2 1 topopn ( 𝐽 ∈ Top → 𝑋𝐽 )
3 pwexg ( 𝑋𝐽 → 𝒫 𝑋 ∈ V )
4 mptexg ( 𝒫 𝑋 ∈ V → ( 𝑥 ∈ 𝒫 𝑋 { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } ) ∈ V )
5 2 3 4 3syl ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } ) ∈ 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 ) ∣ 𝑥𝑦 } = { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } )
11 10 inteqd ( 𝑗 = 𝐽 { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥𝑦 } = { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } )
12 8 11 mpteq12dv ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 𝑗 { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥𝑦 } ) = ( 𝑥 ∈ 𝒫 𝑋 { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } ) )
13 df-cls cls = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 𝑗 { 𝑦 ∈ ( Clsd ‘ 𝑗 ) ∣ 𝑥𝑦 } ) )
14 12 13 fvmptg ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } ) ∈ V ) → ( cls ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } ) )
15 5 14 mpdan ( 𝐽 ∈ Top → ( cls ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 { 𝑦 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑥𝑦 } ) )