Image of Preimage under Mapping/Corollary

Corollary to Image of Preimage of Mapping
Let $f: S \to T$ be a mapping.

Then:
 * $B \subseteq \operatorname{Im} \left({S}\right) \implies \left({f \circ f^{-1}}\right) \left({B}\right) = B$

Proof
From Subset of Image we have:
 * $B \subseteq \operatorname{Im} \left({S}\right) \implies f^{-1} \left({B}\right) \subseteq f^{-1} \left({\operatorname{Im} \left({S}\right)}\right)$

and from Intersection with Subset is Subset we have:


 * $f^{-1} \left({B}\right) \subseteq f^{-1} \left({\operatorname{Im} \left({S}\right)}\right) \implies f^{-1} \left({B}\right) \cap f^{-1} \left({\operatorname{Im} \left({S}\right)}\right) = f^{-1} \left({B}\right)$

Hence the result.