$|\mathcal{P}(A)|=|\mathcal{P}(B)|$ implies that $|A|=|B|$?

Question

I know that if $|A|=|B|$, then we can extend the function such that $|\mathcal{P}(A)|=|\mathcal{P}(B)|$ but can the bijective function $f:P(A)\to P(B)$ somehow be used to give a bijective from $A$ to $B$ or is it not possible? And the proposition would not be true