Is the following correct
$$E[E[\mathbb{I}(X)]] = E[\mathbb{I}(X)]$$
I assume that $E[E[X]] = E[X]$, as $E[X]$ is a number and expected value of a constant is a constant, and that $\mathbb{I}(X)$ has a binomial distribution. $\mathbb{I}(X)$ represents an indicator function of some random variable $X$.
Exact definition of $\mathbb{I}()$ is not important.
Yes, $E[f(X)]$ is nonrandom, so $E[E[f(X)]] = E[f(X)]$ for any function $f$ for which the terms are well-defined.