# Composite of Bijections is Bijection/Proof 1

## Theorem

Let $f$ and $g$ be mappings such that $\Dom f = \Cdm g$.

Then:

If $f$ and $g$ are both bijections, then so is $f \circ g$

where $f \circ g$ is the composite mapping of $f$ with $g$.

## Proof

As every bijection is also by definition an injection, a composite of bijections is also a composite of injections.

Every composite of injections is also an injection by Composite of Injections is Injection.

As every bijection is also by definition a surjection, a composite of bijections is also a composite of surjections.

Every composite of surjections is also a surjection by Composite of Surjections is Surjection.

As a composite of bijections is therefore both an injection and a surjection, it is also a bijection.

$\blacksquare$