Suppose we have a Markov chain $X=\{X_0, X_1, ...\},$ with state space $S$, and a subset of states $A\subseteq S$.
Define $T_A:=\min \{n\geq 0: X_n\in A\}$.
Define notations $\,E_x[Y]:=E[Y|X_0 =x]$ for all $x\in S$ $\quad$ and $\quad$ $P_x(X_1=y):=P(X_1=y|X_0=x).$
Now for $x\notin A$, we calculate
$g(x)=E_x[T_A]=\sum_{y\in S} E_x[T_A 1_{\{X_1=y\}}]=1+ \sum_{y\in S} E_x[(T_A -1) 1_{\{X_1=y\}}].$
$\color{red}{\text{My teacher then wrote that}}$ $$\sum_{y\in S} E_x[(T_A -1) 1_{\{X_1=y\}}] \color{red}{=} \sum_{y\in S} P_x(X_1=y) E_x\big[ E_x[(T_A -1)|X_1=y]\big]$$ and $$E_x\big[ E_x[(T_A -1)|X_1=y]\big]\color{red}{=}E_y[T_A].$$
I guess he has applied Tower property twice to derive the two equalities above (in $\color{red}{\text{red}}$) but I don't know how he did so. The problem is that, he is conditioning on a set $[X_1=y]$ rather than on a random variable, say $X_1$. How can he applies Tower property here?
My question: how did he derive the two equalities above (in $\color{red}{\text{red}}$)? Thanks for any help.