Exercise C.3.5

$\star$ Let $X$ and $Y$ be independent random variables. Prove that $f(X)$ and $g(Y)$ are independent for any choice of functions $f$ and $g$.

(UNSOLVED) This is so intuitively obvious, that I have a hard time wanting to do it. On the other hand, intuition disagrees with many things in probability thoery. Either way, I have no clue what to do with $\Pr\{f(X)\}$, so I don't know how to prove it.