functions - Images and preimages of sets

This is probably very basic indeed, but I've just ran into a proof that seems to imply that letting $f:X\to Y$, $C\subseteq X$ and $P\subseteq Y$, $Q\subseteq Y$,

$$ f(C)=P\cup Q\iff C=f^{-1}(P\cup Q),$$

which I can only understand to be true if f is bijective. Am I having a mental blackout?

Thanks,

Miguel