Canonical morphim_group fPh G := @group _ (morphim fPh G) (morphim_groupset G).
Canonical ker_group fPh : {group aT} := Eval hnf in [group of ker fPh].

Lemma sub_morphim_pre A R :
A \subset D -> (f @* A \subset R) = (A \subset f @*^-1 R).