Theorem ntrfval

Description: The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)

	Ref	Expression
Hypothesis	cldval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	ntrfval	⊢ ( 𝐽 ∈ Top → ( int ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) )

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 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) ∈ V )
5	2 3 4	3syl	⊢ ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) ∈ 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		ineq1	⊢ ( 𝑗 = 𝐽 → ( 𝑗 ∩ 𝒫 𝑥 ) = ( 𝐽 ∩ 𝒫 𝑥 ) )
10	9	unieqd	⊢ ( 𝑗 = 𝐽 → ∪ ( 𝑗 ∩ 𝒫 𝑥 ) = ∪ ( 𝐽 ∩ 𝒫 𝑥 ) )
11	8 10	mpteq12dv	⊢ ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ ( 𝑗 ∩ 𝒫 𝑥 ) ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) )
12		df-ntr	⊢ int = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ ( 𝑗 ∩ 𝒫 𝑥 ) ) )
13	11 12	fvmptg	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) ∈ V ) → ( int ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) )
14	5 13	mpdan	⊢ ( 𝐽 ∈ Top → ( int ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ ∪ ( 𝐽 ∩ 𝒫 𝑥 ) ) )