∀ {ι : Type u_1} {X : Type u_2} {α : Type u_3} [inst : TopologicalSpace X] [inst_1 : UniformSpace α] {F : ι → X → α}
  [inst_2 : UniformSpace ι] {𝔖 : Set (Set X)},
  ⋃₀ 𝔖 = Set.univ →
    (∀ K ∈ 𝔖, IsCompact K) →
      (∀ K ∈ 𝔖, EquicontinuousOn F K) → (IsUniformInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ↔ IsUniformInducing F)