Image of Null is Null

Theorem

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

The image of the empty set is the empty set:

$\mathcal R \left({\varnothing}\right) = \varnothing$

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \mathcal R \left({\varnothing}\right) = \left\{ {t \in \operatorname{Rng} \left({\mathcal R}\right): \exists s \in \varnothing: \left({s, t}\right) \in \mathcal R}\right\}\)	\(\displaystyle \)	Definition of image of subset
	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \neg \exists s \in \varnothing\)	\(\displaystyle \)	Definition of empty set
	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \neg \exists t \in \left\{ {t \in \operatorname{Rng} \left({\mathcal R}\right): \exists s \in \varnothing: \left({s, t}\right) \in \mathcal R}\right\}\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \mathcal R \left({\varnothing}\right) = \varnothing\)	\(\displaystyle \)	Definition of empty set

$\blacksquare$