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 } ) )