Direct Image Mapping is Bijection iff Mapping is Bijection

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

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

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