Direct Image Mapping is Bijection iff Mapping is Bijection

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

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

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