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 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) )

Proof

Step Hyp Ref Expression
1 neifval.1 𝑋 = 𝐽
2 1 topopn ( 𝐽 ∈ Top → 𝑋𝐽 )
3 pwexg ( 𝑋𝐽 → 𝒫 𝑋 ∈ V )
4 mptexg ( 𝒫 𝑋 ∈ V → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) ∈ V )
5 2 3 4 3syl ( 𝐽 ∈ Top → ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) ∈ 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 → ( ∃ 𝑔𝑗 ( 𝑥𝑔𝑔𝑣 ) ↔ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) ) )
10 8 9 rabeqbidv ( 𝑗 = 𝐽 → { 𝑣 ∈ 𝒫 𝑗 ∣ ∃ 𝑔𝑗 ( 𝑥𝑔𝑔𝑣 ) } = { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } )
11 8 10 mpteq12dv ( 𝑗 = 𝐽 → ( 𝑥 ∈ 𝒫 𝑗 ↦ { 𝑣 ∈ 𝒫 𝑗 ∣ ∃ 𝑔𝑗 ( 𝑥𝑔𝑔𝑣 ) } ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) )
12 df-nei nei = ( 𝑗 ∈ Top ↦ ( 𝑥 ∈ 𝒫 𝑗 ↦ { 𝑣 ∈ 𝒫 𝑗 ∣ ∃ 𝑔𝑗 ( 𝑥𝑔𝑔𝑣 ) } ) )
13 11 12 fvmptg ( ( 𝐽 ∈ Top ∧ ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) ∈ V ) → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) )
14 5 13 mpdan ( 𝐽 ∈ Top → ( nei ‘ 𝐽 ) = ( 𝑥 ∈ 𝒫 𝑋 ↦ { 𝑣 ∈ 𝒫 𝑋 ∣ ∃ 𝑔𝐽 ( 𝑥𝑔𝑔𝑣 ) } ) )