Image of Singleton under Relation

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:

So $R \left({s}\right) \subseteq \mathcal R \left[{\left\{ {s}\right\} }\right]$:

And $\mathcal R \left[{\left\{ {s}\right\} }\right] \subseteq R \left({s}\right)$:

Finally $\mathcal R \left({s}\right) = \mathcal R \left[{\left\{{s}\right\}}\right]$ by Definition:Set_Equality.