Subset of Codomain is Superset of Image of Preimage/Proof 2

Proof
Let $y \in B$.

Then:
 * $\exists x \in S: y = f \left({x}\right)$

Therefore by definition of preimage of subset:
 * $\exists x \in f^{-1} \left[{B}\right]$

It follows by definition of image of subset that:
 * $y \in f \left[{f^{-1} \left[{B}\right]}\right]$

Thus by definition of composition $f$ with $f^{-1}$:
 * $y \in \left({f \circ f^{-1}}\right) \left[{B}\right]$

The result follows by definition of subset.