∀ {X : Type u_1} {𝕜 : Type u_2} {A : Type u_3} {p : A → Prop} [inst : RCLike 𝕜] [inst_1 : NormedRing A]
  [inst_2 : StarRing A] [inst_3 : NormedAlgebra 𝕜 A] [inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p]
  [ContinuousStar A] {s : Set 𝕜},
  IsCompact s →
    ∀ (f : 𝕜 → 𝕜) {a : X → A} {a₀ : A} {l : Filter X},
      Filter.Tendsto a l (nhds a₀) →
        (∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s) →
          (∀ᶠ (x : X) in l, p (a x)) →
            spectrum 𝕜 a₀ ⊆ s →
              p a₀ →
                autoParam (ContinuousOn f s) Filter.Tendsto.cfc._auto_1 →
                  Filter.Tendsto (fun x => cfc f (a x)) l (nhds (cfc f a₀))