∀ {α : Type u_1} {β : Type u_2} {ι : Type u_4} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] {F : ι → α → β}
  {f : α → β} {p : Filter ι} (S : Set (Set α)),
  (∀ s ∈ S, IsOpen s) → (∀ s ∈ S, TendstoLocallyUniformlyOn F f p s) → TendstoLocallyUniformlyOn F f p (⋃₀ S)