∀ {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [inst : NormedField 𝕜₁] [inst_1 : NormedField 𝕜₂] {σ : 𝕜₁ →+* 𝕜₂} {E : Type u_4}
  {F : Type u_5} [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜₁ E] [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜₂ F]
  [inst_6 : TopologicalSpace E] [inst_7 : TopologicalSpace F] [inst_8 : IsTopologicalAddGroup F] {ι : Type u_7}
  {p : ι → Prop} {b : ι → Set F},
  (nhds 0).HasBasis p b →
    (nhds 0).HasBasis (fun Si => Bornology.IsVonNBounded 𝕜₁ Si.1 ∧ p Si.2) fun Si => {f | ∀ x ∈ Si.1, f x ∈ b Si.2}