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Metamath Proof Explorer

Theorem neif

Description: The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis neifval.1 ⊢ X = ⋃ J
Assertion neif ⊢ J ∈ Top → nei ⁡ J Fn 𝒫 X

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 rabexg ⊢ 𝒫 X ∈ V → v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v ∈ V
5 2 3 4 3syl ⊢ J ∈ Top → v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v ∈ V
6 5 ralrimivw ⊢ J ∈ Top → ∀ x ∈ 𝒫 X v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v ∈ V
7 eqid ⊢ x ∈ 𝒫 X ⟼ v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v = x ∈ 𝒫 X ⟼ v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v
8 7 fnmpt ⊢ ∀ x ∈ 𝒫 X v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v ∈ V → x ∈ 𝒫 X ⟼ v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v Fn 𝒫 X
9 6 8 syl ⊢ J ∈ Top → x ∈ 𝒫 X ⟼ v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v Fn 𝒫 X
10 1 neifval ⊢ J ∈ Top → nei ⁡ J = x ∈ 𝒫 X ⟼ v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v
11 10 fneq1d ⊢ J ∈ Top → nei ⁡ J Fn 𝒫 X ↔ x ∈ 𝒫 X ⟼ v ∈ 𝒫 X | ∃ g ∈ J x ⊆ g ∧ g ⊆ v Fn 𝒫 X
12 9 11 mpbird ⊢ J ∈ Top → nei ⁡ J Fn 𝒫 X