Theorem gneispacess

Description: All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (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	gneispacess	⊢ ( 𝐹 ∈ 𝐴 → ∀ 𝑝 ∈ dom 𝐹 ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 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	gneispace2	⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 ∈ 𝐴 ↔ ( 𝐹 : dom 𝐹 ⟶ ( 𝒫 ( 𝒫 dom 𝐹 ∖ { ∅ } ) ∖ { ∅ } ) ∧ ∀ 𝑝 ∈ dom 𝐹 ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) )
3	2	ibi	⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 : dom 𝐹 ⟶ ( 𝒫 ( 𝒫 dom 𝐹 ∖ { ∅ } ) ∖ { ∅ } ) ∧ ∀ 𝑝 ∈ dom 𝐹 ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) )
4		simpr	⊢ ( ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) → ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) )
5	4	2ralimi	⊢ ( ∀ 𝑝 ∈ dom 𝐹 ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) → ∀ 𝑝 ∈ dom 𝐹 ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) )
6	3 5	simpl2im	⊢ ( 𝐹 ∈ 𝐴 → ∀ 𝑝 ∈ dom 𝐹 ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) )