Subset equals Preimage of Image implies Injection/Proof 1

Proof
Let $f$ be such that:
 * $\forall A \in \mathcal P \left({S}\right): A = \left({f^\gets \circ f^\to}\right) \left({A}\right)$

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.