Let $X$ be a complex irreducible quasi-projective variety, $f:X\longrightarrow\mathbb{P}^N$ a morphism, $H\subset\mathbb{P}^N$ a hyperplane, $Z:=f^{-1}(H)$ which is irreducible, $Y\subset X$ a irriducible closed subset. Clearly we have $f(Y)\cap H=f(Y\cap Z)$, is it true that $\overline{f(Y)}\cap H=\overline{f(Y\cap Z)}$?

EDIT: Moreover $Y\cap Z$ is irreducible (hence not empty).

Thanks.