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Metamath Proof Explorer

Theorem neifval

Description: Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis neifval.1 ⊢ 𝑋 = ∪ 𝐽
Assertion neifval ( 𝐽 ∈ Top → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) )

Proof

Step Hyp Ref Expression
1 neifval.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 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ∈ V )
5 2 3 4 3syl ⊢ ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ∈ 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 rexeq ⊢ ( 𝑗 = 𝐽 → ( ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ↔ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) ) )
10 8 9 rabeqbidv ⊢ ( 𝑗 = 𝐽 → { 𝑣 ∈ 𝒫 ∪ 𝑗 ∣ ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } )
11 8 10 mpteq12dv ⊢ ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ { 𝑣 ∈ 𝒫 ∪ 𝑗 ∣ ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) )
12 df-nei ⊢ nei = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 ∪ 𝑗 ↦ { 𝑣 ∈ 𝒫 ∪ 𝑗 ∣ ∃ 𝑔 ∈ 𝑗 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) )
13 11 12 fvmptg ⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) ∈ V ) → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) )
14 5 13 mpdan ⊢ ( 𝐽 ∈ Top → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔 ∈ 𝐽 ( 𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣 ) } ) )