Mapping to Singleton is Unique
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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.
$\blacksquare$