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Metamath Proof Explorer

Theorem cldval

Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis cldval.1 ⊢ 𝑋 = ∪ 𝐽
Assertion cldval ( 𝐽 ∈ Top → ( Clsd ‘ 𝐽 ) = { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } )

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 rabexg ⊢ ( 𝒫 𝑋 ∈ V → { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } ∈ V )
5 2 3 4 3syl ⊢ ( 𝐽 ∈ Top → { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } ∈ 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 7 difeq1d ⊢ ( 𝑗 = 𝐽 → ( ∪ 𝑗 ∖ 𝑥 ) = ( 𝑋 ∖ 𝑥 ) )
10 eleq12 ⊢ ( ( ( ∪ 𝑗 ∖ 𝑥 ) = ( 𝑋 ∖ 𝑥 ) ∧ 𝑗 = 𝐽 ) → ( ( ∪ 𝑗 ∖ 𝑥 ) ∈ 𝑗 ↔ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 ) )
11 9 10 mpancom ⊢ ( 𝑗 = 𝐽 → ( ( ∪ 𝑗 ∖ 𝑥 ) ∈ 𝑗 ↔ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 ) )
12 8 11 rabeqbidv ⊢ ( 𝑗 = 𝐽 → { 𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ( ∪ 𝑗 ∖ 𝑥 ) ∈ 𝑗 } = { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } )
13 df-cld ⊢ Clsd = ( 𝑗 ∈ Top ↦ { 𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ( ∪ 𝑗 ∖ 𝑥 ) ∈ 𝑗 } )
14 12 13 fvmptg ⊢ ( ( 𝐽 ∈ Top ∧ { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } ∈ V ) → ( Clsd ‘ 𝐽 ) = { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } )
15 5 14 mpdan ⊢ ( 𝐽 ∈ Top → ( Clsd ‘ 𝐽 ) = { 𝑥 ∈ 𝒫 𝑋 ∣ ( 𝑋 ∖ 𝑥 ) ∈ 𝐽 } )