∀ {α : Type u_1} {ι : Sort u_2} {β : Sort u_3} (S : ι → Set α) (f f_1 : (i : ι) → ↑(S i) → β) (e_f : f = f_1)
  (x : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩) (T : Set α)
  (hT : T ⊆ Set.iUnion S) (x_1 x_2 : ↑T), x_1 = x_2 → Set.iUnionLift S f x T hT x_1 = Set.iUnionLift S f_1 ⋯ T hT x_2