Composite of Injections is Injection

Theorem

A composite of injections is an injection.

That is:

If $f$ and $g$ are injections, then so is $f \circ g$.

Proof

Let $f$ and $g$ be injections.

Then:

 $\ds \map {f \circ g} x$ $=$ $\ds \map {f \circ g} y$ $\ds \leadsto \ \$ $\ds \map f {\map g x}$ $=$ $\ds \map f {\map g y}$ Definition of Composition of Mappings $\ds \leadsto \ \$ $\ds \map g x$ $=$ $\ds \map g y$ as $f$ is injective $\ds \leadsto \ \$ $\ds x$ $=$ $\ds y$ as $g$ is injective

$\blacksquare$