Let $X$ and $Y$ be sets. It is undecidable in ZFC whether $2^{|X|} = 2^{|Y|}$ implies $|X| = |Y|$ (in Cohen's original model for ZFC + $\neg$CH, one has $2^{\aleph_0} = 2^{\aleph_1}$).

Question. Do $X$ and $Y$ have the same cardinality if the families of finite subsets of both sets do?

I feel this must be either basic or very well known (to those who know it very well).