I would like to show that there is a bijection from $\mathcal{P}(X \times Y)$ to $(X \to \mathcal{P}(Y))$ and prove the opposite relationship.
That is, show that $\mathcal{P}(X \times Y) \cong(X \to \mathcal{P}(Y))$ and $ (X \to \mathcal{P}(Y)) \cong \mathcal{P}(X \times Y)$. I know some relationships like $\mathcal{P}(X) = (X \implies [2])$ (and there are many more) but I can't seem to link it to the bijection relationships as required.