TITLE: Prove $F(F^{-1}(B)) = B$ for onto function
QUESTION [9 upvotes]: Suppose that $f:X \to Y$ is an onto function. Prove that for all subsets $B$ subset of $Y$, $f(f^{-1}(B)) = B$. I don't know how to do this if the function is not also one to one, which it is not. Any help proving this would be greatly appreciated.

REPLY [6 votes]: $\exists$ is the symbol for "there exists."
$\wedge$ is the symbol for "and."
$f:X\twoheadrightarrow Y$ is notation for "$f$ is a surjective (onto) function from $X$ to $Y$."
$\mathcal{P}(Y)$ is notation for "powerset of $Y$" or "the set of all subsets of $Y$."



"$\subseteq$" becomes "$=$" for onto functions.