# Image is Subset of Codomain/Corollary 1

## Corollary to Image is Subset of Codomain

Let $\mathcal R = S \times T$ be a relation.

The image of $\mathcal R$ is a subset of the codomain of $\mathcal R$:

- $\Img {\mathcal R} \subseteq T$

## Proof

\(\displaystyle \Img {\mathcal R}\) | \(=\) | \(\displaystyle \mathcal R \sqbrk {\Dom {\mathcal R} }\) | $\quad$ Definition of Image of Relation | $\quad$ | |||||||||

\(\displaystyle \Dom {\mathcal R}\) | \(\subseteq\) | \(\displaystyle \Dom {\mathcal R}\) | $\quad$ Set is Subset of Itself | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \Img {\mathcal R}\) | \(\subseteq\) | \(\displaystyle T\) | $\quad$ Image is Subset of Codomain | $\quad$ |

$\blacksquare$