Image is Subset of Codomain/Corollary 1
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Corollary to Image is Subset of Codomain
Let $\RR = S \times T$ be a relation.
The image of $\RR$ is a subset of the codomain of $\RR$:
- $\Img \RR \subseteq T$
Proof
| \(\ds \Img \RR\) | \(=\) | \(\ds \RR \sqbrk {\Dom \RR}\) | Definition of Image of Relation | |||||||||||
| \(\ds \Dom \RR\) | \(\subseteq\) | \(\ds \Dom \RR\) | Set is Subset of Itself | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \Img \RR\) | \(\subseteq\) | \(\ds T\) | Image is Subset of Codomain |
$\blacksquare$