∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_6} {β : Type u_8} [uα : UniformSpace α] [uβ : UniformSpace β] {p : κ → Prop}
  {s : κ → Set (β × β)} {F : ι → β → α} {S : Set β},
  (uniformity β ⊓ Filter.principal (S ×ˢ S)).HasBasis p s →
    (UniformEquicontinuousOn F S ↔
      ∀ U ∈ uniformity α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U)