Subset equals Preimage of Image iff Mapping is Injection

Theorem
Let $g: S \to T$ be a mapping.

Let $f_g: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ be the mapping induced by $g$.

Similarly, let $f_{g^{-1}}: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right)$ be the mapping induced by the inverse $g^{-1}$.

Then:
 * $\forall A \in \mathcal P \left({S}\right): A = \left({f_{g^{-1}} \circ f_g}\right) \left({A}\right)$

iff $g$ is an injection.

Sufficient Condition
Let $g$ be such that:
 * $\forall A \in \mathcal P \left({S}\right): A = \left({f_{g^{-1}} \circ f_g}\right) \left({A}\right)$

Then by Subset equals Preimage of Image implies Injection, $g$ is an injection.

Necessary Condition
Let $g$ be an injection.

Then by Preimage of Image of Subset under Injection equals Subset:


 * $\forall A \in \mathcal P \left({S}\right): A = \left({f_{g^{-1}} \circ f_g}\right) \left({A}\right)$