Subset_equals_Image_of_Preimage_iff_Mapping_is_Surjection

Let $f: S \to T$ be a mapping. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$. Similarly, let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$. Then $\forall B \in \powerset T: B = \map {\paren {f^\to \circ f^\gets} } B$ $f$ is a surjection.