# Mapping from Singleton is Injection

## Theorem

Let $f: S \to T$ be a mapping.

Let $S$ be a singleton.

Then $f$ is an injection.

## Proof

Let $S = \set s$.

For $f$ to be an injection, all we need to do is show:

$\forall x_1, x_2 \in S: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$

But as $S$ is a singleton, it follows that:

$x_1 = x_2 = s$

Hence the result.

$\blacksquare$