Image of Composite Mapping/Corollary

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

Then:


 * $\Img {f \circ g} \subseteq \Img f$

where:
 * $f \circ g$ denotes composition of $g$ and $f$
 * $\Img f$ denotes image of $f$.

Proof
From Image of Composite Mapping, it holds that:


 * $\Img {f \circ g} = f \sqbrk {\Img g}$

where $f \sqbrk {\, \cdot \,}$ denotes image of subset.

By definition of composite mapping:
 * $\Img g \subseteq \Dom f$

where $\Dom f$ denotes the domain of $f$.

From Image of Subset under Mapping is Subset of Image:


 * $\Img {f \circ g} \subseteq \Img f$