Empty Mapping is Unique

Theorem
For each set $T$ there exists exactly one empty mapping $f: \O \to T$ whose domain is the empty set.

Proof
By definition a mapping from $\O$ to $T$ is a subset of the cartesian product $\O \times T$:
 * $f: \O \to T \subseteq \O \times T$

From Empty Mapping is Mapping, we have that the empty mapping from $\O$ to $T$ exists.

From Cartesian Product is Empty iff Factor is Empty it follows that the empty mapping equals the empty set:
 * $\O \times T = \O$

The result follows by Empty Set is Unique.