lean_workbook_plus_28710

Let $P$ and $Q$ be sets and $f$ be a function from $P$ to $Q$. For every subset $X\subseteq Q$, define $f^{-1}(X)=\{x\in P \mid f(x)\in X\}$. Show that $f(f^{-1}(B))=B$ if and only if $f$ is onto. This has to be taken with a universal quantifier for all $B\subseteq Q$.