Breaking up Expectation over a Joint Distribution

Question

For some distribution $D\in \Delta(\mathcal{X}\times \mathcal{Y} )$ and function $f : \mathcal{X}\times \mathcal{Y} \to \mathbb{R}$, is it true that: \begin{align*} \mathbb{E}_{(x,y)\sim D} [f(x,y)] = \mathbb{E}_{x} [\mathbb{E}_y[f(x,y)\, |\, x]]. \end{align*}

Answer 1

Yes, this is the Law of Total Expectation.