The problem of describing the solutions of a polynomial system appears in many different fields such as robotic, control theory, etc. When the system depends on parameters, its minimal discriminant variety is the set of parameter values around which the roots of the system cannot be expressed as a continuous function of the parameters. In particular, an important component of the minimal discriminant variety is the set of properness defects. This article presents a method efficient in practice and in theory to compute the non-properness set of a projection mapping, by reducing the problem to a problem of variable elimination. We also present a reduction of the computation of the minimal discriminant variety to the computation of the non-properness set of a projection mapping. This result allows us to deduce a bound on the degree and the time computation of the minimal discriminant variety of a parametric system under some assumptions.