(** Church's iota operator *)
Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (classical_definite_description P i).
Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(exists! x:A, P x) -> P (iota i P)
:= proj2_sig (classical_definite_description P i).

Theorem unique_choice :
forall (A B:Type) (R:A -> B -> Prop),
(forall x:A,  exists! y : B, R x y) ->
(exists f : A -> B, forall x:A, R x (f x)).