The connection between fluid dynamics and classical statistical mechanics has motivated in the past mathematical investigations of the incompressible Navier-Stokes (NS) equations (INSE) by means of an asymptotic kinetic theory. This feature has suggestedthe search for possible alternative exact approaches, based on the construction of a suitable inverse kinetic theory (IKT), which can avoid the asymptotic character and the intrinsic mathematical difficulty of direct kinetic theories. In this paper thefundamental mathematical properties of the NS phase-space dynamical system underlying INSE and determined by IKT are investigated. In particular, an equivalence theorem with the INSE problem and a global existence theorem are proved to hold for the NS dynamical system.