This paper is the ninth in a sequence on the structure of sets of solutions
to systems of equations in free and hyperbolic groups, projections of such sets
(Diophantine sets), and the structure of definable sets over free and
hyperbolic groups. In the ninth paper we associate a Diophantine set with a
definable set, and view it as the Diophantine envelope of the definable set. We
use the envelope and duo limit groups that were used in proving stability of
the theory of free and torsion-free hyperbolic groups [Se9], to study definable
equivalence relations, and in particular, to classify imaginaries over these
groups.