Theorem gneispace0nelrn2

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	gneispace0nelrn2	⊢ ( ( 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	1	gneispace0nelrn	⊢ ( 𝐹 ∈ 𝐴 → ∀ 𝑝 ∈ dom 𝐹 ( 𝐹 ‘ 𝑝 ) ≠ ∅ )
3		fveq2	⊢ ( 𝑝 = 𝑃 → ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑃 ) )
4	3	neeq1d	⊢ ( 𝑝 = 𝑃 → ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ↔ ( 𝐹 ‘ 𝑃 ) ≠ ∅ ) )
5	4	rspccv	⊢ ( ∀ 𝑝 ∈ dom 𝐹 ( 𝐹 ‘ 𝑝 ) ≠ ∅ → ( 𝑃 ∈ dom 𝐹 → ( 𝐹 ‘ 𝑃 ) ≠ ∅ ) )
6	2 5	syl	⊢ ( 𝐹 ∈ 𝐴 → ( 𝑃 ∈ dom 𝐹 → ( 𝐹 ‘ 𝑃 ) ≠ ∅ ) )
7	6	imp	⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑃 ) ≠ ∅ )