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:

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

$\blacksquare$

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.14$: Composition of Functions