Direct Image Mapping is Bijection iff Mapping is Bijection

Theorem
Let $$\mathcal R \subseteq S \times T$$ be a relation.

Let $$f_{\mathcal R}: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$$ be the mapping induced on $\mathcal P \left({S}\right)$ by $\mathcal R$.

Then $$\mathcal R \subseteq S \times T$$ is a bijection iff $$f_{\mathcal R}: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$$ is a bijection.