# Image of Empty Set is Empty Set

## Theorem

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

The image of the empty set is the empty set:

$\mathcal R \sqbrk \O = \O$

### Corollary

Let $f: S \to T$ be a mapping.

The image of the empty set is the empty set:

$f \sqbrk \O = \O$

## Proof

 $\displaystyle \mathcal R \sqbrk \O$ $=$ $\displaystyle \set {t \in \Rng {\mathcal R}: \exists s \in \O: \tuple {s, t} \in \mathcal R}$ $\quad$ Definition of Image of Subset under Relation $\quad$ $\displaystyle \neg \exists s$ $\in$ $\displaystyle \O$ $\quad$ Definition of Empty Set $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \neg \exists t$ $\in$ $\displaystyle \set {t \in \Rng {\mathcal R}: \exists s \in \O: \tuple {s, t} \in \mathcal R}$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \mathcal R \sqbrk \O$ $=$ $\displaystyle \O$ $\quad$ Definition of Empty Set $\quad$

$\blacksquare$