∀ {R : Type u_2} {S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsLocalRing R]
  [inst_3 : Nontrivial S] (f : R →+* S), ↑f.range = ↑(LocalSubring.map f { toSubring := ⊤, isLocalRing := ⋯ }).toSubring