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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \\colon X \\to Y$ be an [[isometry|0878DN3WCS]].\nThen $f$ is [[injective|0PVG1CRAJN]].\n\n## Remarks\n\nProof: if $f(x) = f(y)$, then $d_X(x, y) = d_Y(f(x), f(y)) = 0$, thus $x = y$.\nThe proof uses the axiom of a metric that $d(x, y) = 0$ implies $x = y$. For a pseudometric, the statement can fail.\nThus an isometry is a bijection from $X$ onto its image $f(X)$.\n\n## Explanations\n\n### L1 human\n\nTwo different points have a positive distance, and an isometry keeps that distance.\nSo their images are also at a positive distance, and they are different.\n",
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