∀ {X : Type u_1} [inst : TopologicalSpace X] {ι : Type u_3} {p : X → ι → Prop} {s : X → ι → Set X},
  (∀ (x : X), (nhds x).HasBasis (p x) (s x)) →
    (∀ (x : X) (i : ι), p x i → IsPathConnected (s x i)) → LocPathConnectedSpace X