How to prove an isometry is injective?

Question

So I was given the following statement:

$f:\mathbb E^{2}\longrightarrow \mathbb E^{2}$ is an isometry if and only if $\forall X,Y\in \mathbb E^{2}, d(X,Y)=d(f(X),f(Y))$.

How would I prove that $f$ is an injective?

Answer 1

$f(x)=f(y)$ implies that $d(f(x),f(y))=0=d(x,y)$ and $x=y$ since $d$ is a metric