Proceeding directly from the definitions, we have $f A = \{ f(x) : x \in A \}$ and thus \begin{equation} \begin{split} f^{-1}(f A) & = \{ x \in X : f(x) \in f A \} \\ & = \left\{ x \in X : f(x) \in \{ f(z) : z \in A \} \right\} \\ & = \left\{ x \in X : f(x) \text{ satisfies } x \in A \right\} \\ & = \left\{ x \in X : x \in A \right\} \end{split} \end{equation} $\square$