Image is Subset of Codomain/Corollary 1

From ProofWiki
Jump to: navigation, search

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 $\quad$

$\blacksquare$