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 morphpre_proper R S :
R \subset f @* D -> S \subset f @* D ->
(f @*^-1 R \proper f @*^-1 S) = (R \proper S).