∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
  {P' : Type u_3} [inst_3 : CommRing P'] [inst_4 : Algebra R P'] {I : FractionalIdeal S P} {g : P →ₐ[R] P'} {y : P'},
  (y ∈ FractionalIdeal.map g I) = ∃ x ∈ I, g x = y