Conditional Expectation
Although the CEF is random variable, it's not "defined" pointwise.
\[\begin{align*}
&\mathbb{E}[Y|X] :: \Omega \to \mathcal{R} \\
& \mathcal{I} \ \mathbb{P} \ A \ Y = \mathcal{I} \ \mathbb{P} \ A \ \mathbb{E}[Y|X] \quad \forall \ A \in \sigma(X)
\end{align*}\]
The law of iteration of expectations is part of the definition
\[\begin{align*}
& \mathcal{I} \ \mathbb{P} \ \Omega \ Y = \mathcal{I} \ \mathbb{P} \ \Omega \ \mathbb{E}[Y|X]
\end{align*}\]