Inverse Image Mapping of Injection is Surjection

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

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

Let $f^\gets: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right)$ be the mapping induced by the inverse $f^{-1}$.

Then $f^\gets$ is a surjection.

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

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

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

By Subset equals Preimage of Image iff Mapping is Injection:
 * $f^\gets \left({Y}\right) = X$

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