Image of Composite Mapping/Corollary

Corollary to Image of Composite Mapping
Let $f: S \to T$ and $g: R \to S$ be mappings.

Then:


 * $\operatorname{Im} \left({f \circ g}\right) \subseteq \operatorname{Im} \left({f}\right)$

where $f \circ g$ denotes composition of $g$ and $f$, and $\operatorname{Im}$ denotes image.

Proof
From Image of Composite Mapping, it holds that:


 * $\operatorname{Im} \left({f \circ g}\right) = f \left[{\operatorname{Im} \left({g}\right)}\right]$

where $f \left[{\cdot}\right]$ denotes image of subset.

By definition of composite mapping, $\operatorname{Im} \left({g}\right) \subseteq \operatorname{Dom} \left({f}\right)$, the domain of $f$.

Now yields:


 * $\operatorname{Im} \left({f \circ g}\right) \subseteq \operatorname{Im} \left({f}\right)$