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.