Empty Mapping is Mapping

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

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

Vacuously:
 * $\forall x \in \varnothing: \exists y \in T: \left({x, y}\right) \in f$

thus showing that $f$ is left-total.

Also vacuously:
 * $\forall x \in \varnothing: \forall y_1, y_2 \in T: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$

thus showing that $f$ is functional.

Hence the result, from the definition of a mapping.