By definition, $f(\emptyset) : = \{ f(x) : x \in \emptyset \}$. If instead $f(\emptyset) \neq \emptyset$ would be true, then there would have to be an element $x \in X$ for which $x \in \emptyset$. Since $\emptyset$ contains no elements, this is false for all $x \in X$, whence $f(\emptyset)$ must itself be empty. $\square$
