# Identity Mapping is Injection

## Theorem

On any set $S$, the identity mapping $I_S: S \to S$ is an injection.

## Proof

From the definition of the identity mapping:

$\forall x \in S: \map {I_S} x = x$

So:

$\map {I_S} x = \map {I_S} y \implies x = y$

So by definition $I_S$ is an injection.

$\blacksquare$