"Suppose $f: A \to B$ is onto. Show that $\forall \, Y \subset B, f(f^{-1}(Y)) = Y$.

Here's what I did:

1. Observe that $\forall x \in f^{-1}(Y), f(x) \in Y$. Then $f(f^{-1}(Y)) \subset Y$.

2. Sincefis onto, $Y \subset f(A)$.

This is where I'm stuck. How do I continue step 2 to show that $f(f^{-1}(Y)) \supset Y$? Obviously $f^{-1}(Y) \subset A$, but I don't see how that helps...