Let $V$ be an irreducible variety of dimension $k$ living in $\mathbb{F}^n$ for some algebraically closed field $\mathbb{F}$. Let $\pi: \mathbb{F}^n \to \mathbb{F}^m$ be projection onto the first $m$ coordinates. Suppose the dimension of the Zariski closure of $\pi(V)$ is equal to $j$. Is it true that for every $x \in \pi(V)$, the fiber $\pi^{1}(x) \cap V$ has dimension $kj$? If not, what is a counterexample?
