# Identity Mapping is Surjection

## Theorem

On any set $S$, the identity mapping $I_S: S \to S$ is a surjection.

## Proof

The identity mapping is defined as:

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

Then we have:

 $\ds \forall y \in S: \exists x \in S: x$ $=$ $\ds y$ that is, $y$ itself $\ds \leadsto \ \$ $\ds \forall y \in S: \exists x \in S: \map {I_S} x$ $=$ $\ds y$ Definition of $I_S$

Hence the result.

$\blacksquare$