Isometries are injective

Statement

Let $X$ and $Y$ be metric spaces and let $f \colon X \to Y$ be an isometry. Then $f$ is injective.

Remarks

Proof: if $f(x) = f(y)$, then $d_X(x, y) = d_Y(f(x), f(y)) = 0$, thus $x = y$. The proof uses the axiom of a metric that $d(x, y) = 0$ implies $x = y$. For a pseudometric, the statement can fail. Thus an isometry is a bijection from $X$ onto its image $f(X)$.

Explanations

L1 human

Two different points have a positive distance, and an isometry keeps that distance. So their images are also at a positive distance, and they are different.