Countable Function on Power Set of Sample Space is Discrete Random Variable

Theorem
Let $$\left({\Omega, \Sigma, \Pr}\right)$$ be a probability space such that $$\Sigma$$ is the power set of $$\Omega$$.

Let $$f: \Omega \to \R$$ be a function such that $$\operatorname{Im} \left({f}\right)$$ is countable.

Then $$f$$ is a discrete random variable on $$\left({\Omega, \Sigma, \Pr}\right)$$.

Proof
By definition, $$f^{-1} \left({x}\right) \subseteq \Omega$$.

But then $$f^{-1} \left({x}\right) \in \mathcal P \left({\Omega}\right)$$.

Hence the result.