Empty Mapping is Mapping

Theorem
For each set $T$, the empty mapping, where the domain is the empty set, is a mapping.

Proof
Let $f: \O \to T$ be the empty mapping.

Vacuously:
 * $\forall x \in \O: \exists y \in T: \tuple {x, y} \in f$

thus showing that $f$ is left-total.

Also vacuously:
 * $\forall x \in \O: \forall y_1, y_2 \in T: \tuple {x, y_1} \in f \land \tuple {x, y_2} \in f \implies y_1 = y_2$

thus showing that $f$ is many-to-one.

Hence the result, from the definition of a mapping.