How to show that $E[E[X\mid Y]\mid Y] = E[X\mid Y]$

Question

I'm reading a little proof about how the estimation error (based on the conditional expectation estimator, $E[X\mid Y]$) has zero expectation, and at one point the author used the equality that the conditional expectation of the estimator on $Y$ == the estimator itself, i.e. $E[E[X\mid Y]|Y] = E[X\mid Y]$. Can somebody show me how that's derived?

I'm pretty sure it has to do with the law of iterated expectation...

See textbook screenshot:

Answer 1

The conditional expectation is a function of the conditioning variable: $E(X\mid Y)=g(Y)$

Further, for any function of a random variable, $E(h(Y)\mid Y)=h(Y)$. This might be more obvious to you if we regard the conditioning as a fixed value: $E(h(Y)\mid Y=y)=h(y)$