Section PermIn.
Variables (T : finType) (A : {set T}) (f : T -> T).
Hypotheses (injf : {in A &, injective f}) (sBf : f @: A \subset A).
Definition perm_in := perm perm_in_inj.
Hypothesis Gf : f @* G = G.

Lemma preim_autE A : A \subset G -> aut @^-1: A = f @*^-1 A.