Direct Image Mapping of Surjection is Surjection/Proof 3

Proof
Let $f^\gets$ be the mapping induced by the inverse $f^{-1}$.

Let $X \in \mathcal P \left({T}\right)$.

Let $Y = f^\gets \left({T}\right)$.

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

As such a $Y$ exists for each $X \in \mathcal P \left({T}\right)$, $f^\to$ is surjective.