Direct Image Mapping is Bijection iff Mapping is Bijection

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

Let $\mathcal R^\to: \powerset S \to \powerset T$ be the direct image mapping of $\mathcal R$.

Then $\mathcal R \subseteq S \times T$ is a bijection $\mathcal R^\to: \powerset S \to \powerset T$ is a bijection.