Push-forward of degeneracy classes and ampleness

Let $X$ be a projective variety and $E,F$ vector bundles on $X$. Suppose $g: X \rightarrow Y$ is a surjective map onto another variety $Y$. Let $\phi: E \rightarrow F$ be any vector bundle map and $X_{k}(\phi)$ the $k$'th degeneracy locus of $\phi$. We show that the dimension of $g(X_{k}(\phi))$ is at least equal to

under the hypothesis that $E^{*} \otimes F$ is an ample vector bundle on $X$.