# Empty Mapping is Mapping

## Theorem

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

## Proof

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

$\forall x \in \O: \exists y \in T: \tuple {x, y} \in e$

thus showing that $e$ is left-total.

Also vacuously:

$\forall x \in \O: \forall y_1, y_2 \in T: \tuple {x, y_1} \in e \land \tuple {x, y_2} \in e \implies y_1 = y_2$

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

Hence the result, from the definition of a mapping.

$\blacksquare$