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 ) } ) )