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We study φ-equivalence, and the corresponding concepts of finite determinacy and transversal unfoldings. This equivalence was introduced by L. Favaro and C. Mendes who studied φ - stability. We obtain necessary and sufficient conditions for finite φ - determinacy of map-germs that apply to a special class of infinitesimally stable germs w. Building upon results of Egúsquiza for π - stable paths of mappings, we prove genericity theorems for an auxiliary equivalence relation, namely Ã-- equivalence, and for φ equivalence. The main result shows that the evolution of the singularities from a manifold N into the plane can be realized by preserving a foliation defined by a fixed Morse function φ : N → R. Moreover, the generic bifurcations are classified.