Inverse Image Mapping of Injection is Surjection

Theorem

Let $S$ and $T$ be sets.

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

Let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$.

Then $f^\gets$ is a surjection.

Proof

Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping by $f$.

Let $X \in \powerset S$.

Let $Y = \map {f^\to} X$.

By Subset equals Preimage of Image iff Mapping is Injection:

$\map {f^\gets} Y = X$

As such a $Y$ exists for each $X \in \powerset S$, $f^\gets$ is surjective.

$\blacksquare$