Problem with a Borel set in Polish space

I need to prove the following theorem:

Let $X$ be a complete separable metric space and $E \subset X$ a Borel set. Then there exists a complete separable totally disconnected metric space $Y$ and a continuous map $f$ of $Y$ such that $f$ is one-one and maps $Y$ onto $E$.

I have the theorem proved in case when $E=X$ but I don't know how to cope with the general case.
Thank you for help.