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 } )