Subset equals Preimage of Image implies Injection/Proof 1

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

In particular, it holds for all subsets of $A$ which are singletons.

Now, consider any $x, y \in A$.

We have:

So $f$ is an injection.