Sufficient Statistics of $X$ from $Y$
The function $f$ maps $x$ to the conditional distribution of $Y$ given that $X = x$. This is a "deterministic" function. The expression $f(X)$ is a random variable depending on $X$. When $X = x$, the value of this random variable is $f(X) = f(x)$. In words, the random variable $f(X)$ gives the distribution of $Y$ conditional on the value of $X$.

Given $f(X)$, the variables $X,Y$ become independent. Suppose that $f(X) = \phi$. For each $x$ such that $f(x) = \phi$, the conditional distribution of $Y$ given that $X = x$ is $\phi$, that is, $\Pr[Y = y | X = x] = \phi(y)$. This doesn't depend on the value of $x$ as long as $f(x) = \phi$. In particular, the value of $X$ tells us nothing on $Y$, so the conditional mutual information is zero.