Subset equals Preimage of Image implies Injection/Proof 2

Proof
Suppose that $f$ is not an injection.

Then two elements of $S$ map to the same one element of $T$.

That is:
 * $\exists a_1, a_2 \in S, b \in T: f \left({a_1}\right) = f \left({a_2}\right) = b$

Let $A = \left\{{a_1}\right\}$.

Then:

So by the Rule of Transposition:
 * $\forall A \in \mathcal P \left({S}\right): A = \left({f^\gets \circ f^\to}\right) \left({A}\right)$

implies that $f$ is an injection.