Mapping to Singleton is Unique

Theorem
Let $S$ be a set.

Let $T$ be a singleton.

Then there exists a unique mapping $S \to T$.

Proof
Let $T = \set t$.

Let $f$ and $g$ both be mappings from $S$ to $T$.

From Mapping is Constant iff Image is Singleton:
 * $\forall s \in S: \map f s = t$

and:
 * $\forall s \in S: \map g s = t$

The result follows by Equality of Mappings.

Also see

 * Singleton is Terminal Object of Category of Sets