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 morphimSGK A G :
A \subset D -> 'ker f \subset G -> (f @* A \subset f @* G) = (A \subset G).