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:
 * $\exists a_1, a_2 \in S, b \in T: \map f {a_1} = \map f {a_2} = b$

Let $A = \set {a_1}$.

Then:

So by the Rule of Transposition:
 * $\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$

implies that $f$ is an injection.