If $X,Y$ are random variables, such that $\mathbb E|Y|\lt \infty$, $\mathbb E[Y|X] = m(X)$ where $m$ is some Borel measurable function and $(X_1,Y_1)$ come from same distribution as $(X,Y)$. Is it true that $\mathbb E[Y_1|X_1] = m(X_1)$?
Attempt I managed to prove $$\mathbb E[Y_1] = \mathbb E[Y] = \mathbb E[\mathbb E[Y|X]] = \mathbb E[m(X)] = \mathbb E[m(X_1)]$$ which leads to $$\mathbb E[\mathbb E[Y_1|X_1]] = \mathbb E[m(X_1)],$$ but I am not sure how to proceed.