Let $K$ be a field. Let $V/K$ be an affine variety in $A^m$. Let f be a polynomial map (and hence a "morphism of finite type") $f:V\to A^n$. A theorem of Chevalley's tells us that im(f) is either a variety or "almost" a variety - that is, im(f) is a variety $W$ with perhaps a few varieties of lower dimension cut out from it.

Question: is the degree of $W$ (= Zariski closure of im(f)) bounded solely in terms of m, n, deg(V) and the degree of the polynomials $f_1,\dots ,f_n$ defining f (i.e. $f(\vec{x}) = (f_1(\vec{x}),...,f_n(\vec{x}))$?).

This seems intuitively obvious, but I do not know where to look for a reference. (It's also non-obvious how to adapt the proof of Chevalley's theorem I'm looking at so as to give this.) Does anybody where to look this up and/or how to prove this?