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.

$\blacksquare$


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