Definition prop_in1 P & ph {all1 P} :=
  forall x, in_mem x d1 -> P x.Definition prop_in11 P & ph {all2 P} :=
  forall x y, in_mem x d1 -> in_mem y d2 -> P x y.Definition prop_in2 P & ph {all2 P} :=
  forall x y, in_mem x d1 -> in_mem y d1 -> P x y.Definition prop_in111 P & ph {all3 P} :=
  forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d3 -> P x y z.Definition prop_in12 P & ph {all3 P} :=
  forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d2 -> P x y z.Definition prop_in21 P & ph {all3 P} :=
  forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d2 -> P x y z.Definition prop_in3 P & ph {all3 P} :=
  forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d1 -> P x y z.Definition prop_on1 Pf P & phantom T3 (Pf f) & ph {all1 P} :=
  forall x, in_mem (f x) d2 -> P x.Definition prop_on2 Pf P & phantom T3 (Pf f) & ph {all2 P} :=
  forall x y, in_mem (f x) d2 -> in_mem (f y) d2 -> P x y.Definition inPhantom := Phantom Prop.Definition onPhantom {T} P (x : T) := Phantom Prop (P x).Definition bijective_in aT rT (d : mem_pred aT) (f : aT -> rT) :=
  exists2 g, prop_in1 d (inPhantom (cancel f g))
           & prop_on1 d (Phantom _ (cancel g)) (onPhantom (cancel g) f).Definition bijective_on aT rT (cd : mem_pred rT) (f : aT -> rT) :=
  exists2 g, prop_on1 cd (Phantom _ (cancel f)) (onPhantom (cancel f) g)
           & prop_in1 cd (inPhantom (cancel g f)).Notation "{ 'for' x , P }" := (prop_for x (inPhantom P)) : type_scope.Notation "{ 'in' d , P }" := (prop_in1 (mem d) (inPhantom P)) : type_scope.Notation "{ 'in' d1 & d2 , P }" :=
  (prop_in11 (mem d1) (mem d2) (inPhantom P)) : type_scope.Notation "{ 'in' d & , P }" := (prop_in2 (mem d) (inPhantom P)) : type_scope.Notation "{ 'in' d1 & d2 & d3 , P }" :=
  (prop_in111 (mem d1) (mem d2) (mem d3) (inPhantom P)) : type_scope.Notation "{ 'in' d1 & & d3 , P }" :=
  (prop_in21 (mem d1) (mem d3) (inPhantom P)) : type_scope.Notation "{ 'in' d1 & d2 & , P }" :=
  (prop_in12 (mem d1) (mem d2) (inPhantom P)) : type_scope.Notation "{ 'in' d & & , P }" := (prop_in3 (mem d) (inPhantom P)) : type_scope.Notation "{ 'on' cd , P }" :=
  (prop_on1 (mem cd) (inPhantom P) (inPhantom P)) : type_scope.Notation "{ 'on' cd & , P }" :=
  (prop_on2 (mem cd) (inPhantom P) (inPhantom P)) : type_scope.Notation "{ 'on' cd , P & g }" :=
  (prop_on1 (mem cd) (Phantom (_ -> Prop) P) (onPhantom P g)) : type_scope.Notation "{ 'in' d , 'bijective' f }" := (bijective_in (mem d) f) : type_scope.Notation "{ 'on' cd , 'bijective' f }" :=
  (bijective_on (mem cd) f) : type_scope.

Lemma on1S : (forall x, f x \in D2) -> {on D2, allQ1 f} -> allQ1 f.