Image of Singleton

Theorem

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

Then the image of an element of $S$ is equal to the image of a singleton containing that element, the singleton being a subset of $S$:

$\forall s \in S: \mathcal R \left({s}\right) = \mathcal R \left({\left\{{s}\right\}}\right)$.

Proof

We have the definitions:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathcal R \left({s}\right)\)	\(=\)	\(\displaystyle \left\{ {t \in T: \left({s, t}\right) \in \mathcal R}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of image of element
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathcal R \left({\left\{ {s}\right\} }\right)\)	\(=\)	\(\displaystyle \left\{ {t \in T: \exists s \in \left\{ {s}\right\}: \left({s, t}\right) \in \mathcal R}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of image of subset

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle t \in \mathcal R \left({s}\right)\)	\(\implies\)	\(\displaystyle t \in \mathcal R \left({\left\{ {s}\right\} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of image of element
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle t \in \mathcal R \left({\left\{ {s}\right\} }\right)\)	\(\implies\)	\(\displaystyle t \in \mathcal R \left({s}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of image of subset

$\blacksquare$