[Math] Prove $F(F^{-1}(B)) = B$ for onto function

Question

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.

Answer 1

• $\exists$ 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.