Subset of Codomain is Superset of Image of Preimage

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

Then:
 * $B \subseteq T \implies \left({f^\to \circ f^\gets}\right) \left({B}\right) \subseteq B$

where:


 * $f^\to$ denotes the mapping induced on the power set $\mathcal P \left({S}\right)$ of $S$ by $f$
 * $f^\gets$ denotes the mapping induced on the power set $\mathcal P \left({T}\right)$ of $T$ by the inverse $f^{-1}$
 * $f^\to \circ f^\gets$ denotes composition of $f^\to$ and $f^\gets$.