Inverse Image Mapping of Injection is Surjection

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a injection.

Let $f^*: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right)$ be the mapping induced by the inverse relation of $f$.

Then $f^*$ is a surjection.

Proof
Let $f'': \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ be the mapping induced by $f$.

Let $X \in \mathcal P \left({S}\right)$.

Let $Y = f'' \left({X}\right)$.

By Preimage of Image of Injection, $f^* \left({Y}\right) = X$.

As such a $Y$ exists for each $X \in \mathcal P \left({S}\right)$, $f^*$ is surjective.