Subset equals Preimage of Image implies Injection

Theorem
Let $f: S \to T$ be a mapping.

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

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

Let:
 * $\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$

Then $f$ is an injection.

Also see

 * Preimage of Image of Subset under Injection equals Subset
 * Subset equals Preimage of Image iff Mapping is Injection