∀ {ι : Type u_2} {X : Type u_1} {Y : Type u_3} [inst : TopologicalSpace X] {f : ι → Set X} {g : Y → X} (x : Y)
  (motive : (∃ t ∈ nhds (g x), {i | (f i ∩ t).Nonempty}.Finite) → Prop)
  (x_1 : ∃ t ∈ nhds (g x), {i | (f i ∩ t).Nonempty}.Finite),
  (∀ (s : Set X) (hsx : s ∈ nhds (g x)) (hs : {i | (f i ∩ s).Nonempty}.Finite), motive ⋯) → motive x_1