Image of Preimage of Subset under Surjection equals Subset

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

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

Proof
Let $g$ be a surjection.

Let $B \subseteq T$.

Let $b \in B$.

Then:

From Subset of Codomain is Superset of Image of Preimage, we already have that:
 * $f_g \left({f_{g^{-1}} \left({B}\right)}\right) \subseteq B$

So:
 * $B \subseteq f_g \left({f_{g^{-1}} \left({B}\right)}\right) \subseteq B$

and by definition of set equality:
 * $B = f_g \left({f_{g^{-1}} \left({B}\right)}\right)$

Also see

 * Subset equals Image of Preimage implies Surjection
 * Subset equals Image of Preimage iff Mapping is Surjection