Corollary fun_choice_iff_rel_choice_and_functional_rel_reification :
  forall A B : Type, FunctionalChoice_on A B <->
    RelationalChoice_on A B /\ FunctionalRelReification_on A B.

Lemma subset_types_imp_guarded_rel_choice_iff_rel_choice : ProofIrrelevance -> (GuardedRelationalChoice <-> RelationalChoice).