Metamath Proof Explorer

Theorem gneispace0nelrn

Description: A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021)

Ref Expression
Hypothesis gneispace.a ⊢ A = f | f : dom ⁡ f ⟶ 𝒫 𝒫 dom ⁡ f ∖ ∅ ∖ ∅ ∧ ∀ p ∈ dom ⁡ f ∀ n ∈ f ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ f n ⊆ s → s ∈ f ⁡ p
Assertion gneispace0nelrn ⊢ F ∈ A → ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅

Proof

Step Hyp Ref Expression
1 gneispace.a ⊢ A = f | f : dom ⁡ f ⟶ 𝒫 𝒫 dom ⁡ f ∖ ∅ ∖ ∅ ∧ ∀ p ∈ dom ⁡ f ∀ n ∈ f ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ f n ⊆ s → s ∈ f ⁡ p
2 elex ⊢ F ∈ A → F ∈ V
3 1 gneispace ⊢ F ∈ V → F ∈ A ↔ Fun ⁡ F ∧ ran ⁡ F ⊆ 𝒫 𝒫 dom ⁡ F ∧ ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅ ∧ ∀ n ∈ F ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ F n ⊆ s → s ∈ F ⁡ p
4 2 3 syl ⊢ F ∈ A → F ∈ A ↔ Fun ⁡ F ∧ ran ⁡ F ⊆ 𝒫 𝒫 dom ⁡ F ∧ ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅ ∧ ∀ n ∈ F ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ F n ⊆ s → s ∈ F ⁡ p
5 4 ibi ⊢ F ∈ A → Fun ⁡ F ∧ ran ⁡ F ⊆ 𝒫 𝒫 dom ⁡ F ∧ ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅ ∧ ∀ n ∈ F ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ F n ⊆ s → s ∈ F ⁡ p
6 5 simp3d ⊢ F ∈ A → ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅ ∧ ∀ n ∈ F ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ F n ⊆ s → s ∈ F ⁡ p
7 simpl ⊢ F ⁡ p ≠ ∅ ∧ ∀ n ∈ F ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ F n ⊆ s → s ∈ F ⁡ p → F ⁡ p ≠ ∅
8 7 ralimi ⊢ ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅ ∧ ∀ n ∈ F ⁡ p p ∈ n ∧ ∀ s ∈ 𝒫 dom ⁡ F n ⊆ s → s ∈ F ⁡ p → ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅
9 6 8 syl ⊢ F ∈ A → ∀ p ∈ dom ⁡ F F ⁡ p ≠ ∅