I need to prove the following theorem:

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

I have the theorem proved in case when $\displaystyle E=X$ but I don't know how to cope with the general case.

Thank you for help.