Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible.
Does there exists an irreducible component $X'$ of $X$ such that $f' := f_{|X'}:X'\rightarrow Y$ satisfies $(f')^{-1}(y) = f^{-1}(y)$ for $y\in Y$ a general point?