I guess this is very easy, but after I thought some time about it, I still didn't find the idea.
If I have a continuous map $f$ from topological spaces $X$ to $Y$ where $Y$ is irreducible and $X$ has finitely many irreducible components and $f$ is surjective, does then already exist an irreducible component of $X$ such that the restriction of $f$ to this component is still surjective to $Y$?