Here is the set-up. Suppose that $\left \{ A_\alpha : \alpha \in \Lambda \right \}$ is a collection of subsets of a set X and $\left \{ B_\beta : \beta \in \Psi \right \}$ is a collection of subsets of a set Y. If $f: X\rightarrow Y$ is a function, then
$f(f^{-1}(B_\beta)\cap A_\alpha) = B_\beta \cap f(A_\alpha)\;.$
I'm working to prove that one is a subset of the other to show equality. So far I have proved the forward direction with relative ease but the reverse direction is giving me problems.
My only idea is starting:
Let $x \in B_\beta \cap f(A_\alpha)$, then $x \in B_\beta$ and $x \in f(A_\alpha)$. From here I'm not really sure how to introduce the inverse image of $B_\beta$. Any help would be much appreciated.