Direct Image Mapping of Surjection is Surjection/Proof 3

Proof
Let $f^\gets$ be the inverse image mapping of $f$.

Let $Y \in \powerset T$.

Let $X = \map {f^\gets} Y$.

By Subset equals Image of Preimage iff Mapping is Surjection:
 * $\map {f^\to} X = Y$

As such an $X$ exists for each $Y \in \powerset S$, $f^\to$ is surjective.