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Metamath Proof Explorer

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 ) ) )