x-section of closure of E of first category implies x-section of E nowhere dense

Question

Let $E$ be a subset of first category of product space $X \times Y$. Why is the following true: if $(\bar E)_x \subset Y$ is of first category then it follows that $E_x$ is nowhere dense. $E_x$ denotes the $x$-section of E: the projection of $E \cap (\{x\} \times Y)$ down to $Y$.