case mpr =>
  intro hone
  rw [trivial_subgroup, Membership.mem, instMembershipSet] at hone
  simp at hone
  rw [hone]
  rw [kernel, Membership.mem, instMembershipSet]
  simp
  apply hom_preserve_one
α : Type u_1
β : Type u_2
G : group α
H : group β
hom : group_homomorphism G H
φ : α → β := group_homomorphism.fn G H
hphi : φ = group_homomorphism.fn G H
h : injective (group_homomorphism.fn G H)
x : α
⊢ x ∈ trivial_subgroup G → x ∈ kernel G H hom
no goals