Let $f: X\to Y$ be a regular map of projective varieties that is closed (in the sense that it takes Zariski closed sets to Zariski closed sets). Let $V\subset X$ be a quasiprojective subvariety (i.e. locally closed and irreducible). Is $f(V)$ a quasiprojective subvariety of $Y$?

(I'm aware that under arbitrary regular maps, the image of quasiprojective need only be constructible, but I am assuming the map is closed).

(I doubt that this question is really at the level of Mathoverflow, but I asked it at math.stackexchange and got a small number of upvotes but no answer, so I thought I might try to slyly provoke one of y'all into answering before you close it.)