# 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

 $\displaystyle \Img {\mathcal R}$ $=$ $\displaystyle \mathcal R \sqbrk {\Dom {\mathcal R} }$ Definition of Image of Relation $\displaystyle \Dom {\mathcal R}$ $\subseteq$ $\displaystyle \Dom {\mathcal R}$ Set is Subset of Itself $\displaystyle \leadsto \ \$ $\displaystyle \Img {\mathcal R}$ $\subseteq$ $\displaystyle T$ Image is Subset of Codomain

$\blacksquare$