fun x a => if (x \in R) && (a \inRx: (x \in R); case Ry: (y \in R) => // eqxy; first exact: act_inj eqxy.
  by rewrite -eqxy (acts_act nRA Aa) Rx in Ry.
by rewrite eqxy (acts_act nRA Aa) Ry in Rx.Notation "to \o f" := (comp_action to f) : action_scope.

Section PermAction.

Canonical subact_dom_group := [group of subact_dom].