Image of Composite Mapping/Corollary

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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 $f$.

Now Image of Subset under Relation is Subset of Image: Corollary 2 yields:

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

$\blacksquare$


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