∀ {α : Type u_1} (β : Type u_2) {ι : Type u_4} [inst : UniformSpace β] {δ : ι → Type u_6} (φ : (i : ι) → δ i → α),
  (∃ I, I.Finite ∧ ⋃ i ∈ I, Set.range (φ i) = Set.univ) →
    UniformFun.uniformSpace α β =
      ⨅ i,
        UniformSpace.comap (⇑UniformFun.ofFun ∘ (fun x => x ∘ φ i) ∘ ⇑UniformFun.toFun)
          (UniformFun.uniformSpace (δ i) β)