Composite of Bijections is Bijection

Theorem
Every composite of bijections is also a bijection.

That is:
 * If $f$ and $g$ are both bijections, then so is $f \circ g$.

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.