∀ {α : Sort u} {β : α → Sort v} {α' : Sort w} [inst : DecidableEq α] [inst_1 : DecidableEq α'] (g : (a : α) → β a)
  {f : α' → α},
  Function.Injective f →
    ∀ (i : α') (a : β (f i)), (fun j => Function.update g (f i) a (f j)) = Function.update (fun i => g (f i)) i a