Null Relation is Mapping iff Domain is Empty Set

Theorem
Let $S$ and $T$ be sets.

The null relation $\mathcal R = \varnothing \subseteq S \times T$ is a mapping iff $S = \varnothing$.

Sufficient Condition
Let $S = \varnothing$.

Then the null relation $\mathcal R = \varnothing \subseteq S \times T$ is a mapping from Empty Mapping is Mapping.

Necessary Condition
Suppose $S \ne \varnothing$.

From the definition of an empty set, $S \ne \varnothing \implies \exists x \in S$.

Thus: