Relation to Empty Set is Mapping iff Domain is Empty

Theorem

Let $S$ be a set

Let $S \times \O$ denote the cartesian product of $S$ with the empty set $\O$.

Let $\RR \subseteq S \times \O$ be a relation in $S$ to $\O$.

Then $\RR$ is a mapping if and only if $S = \O$.

Proof

Let $S \ne \O$.

Then $\exists s \in S$.

But there exists no $t \in \O$.

Hence there is no $\tuple {s, t} \in \RR$.

So $\RR$ is not a mapping by definition.

Let $S = \O$.

Then $\RR$ is the empty mapping by definition.

From Empty Mapping is Mapping, it is demonstrated that $\RR$ is indeed a mapping.

Hence the result.

$\blacksquare$