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  "body": "\n## Statement\n\nLet $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]].\nThen $f$ is [[bijective|0SNQEF7Z53]] if and only if there is a function $g \\colon B \\to A$ with $g \\circ f = \\mathrm{id}_A$ and $f \\circ g = \\mathrm{id}_B$.\n\n## Remarks\n\nThe function $g$ is then unique. It is the [[inverse function|0W766JAPSN]] $f^{-1}$.\nThis statement does not need the axiom of choice: for a bijection, $g(b)$ is the unique $a$ with $f(a) = b$.\nIf $f$ has a [[left inverse|02486SD328]] $g$ and a [[right inverse|0DDPZAG5ER]] $h$, then $g = h$.\n\n## Explanations\n\n### L1 human\n\nA bijection can be undone in both directions.\nGoing from $A$ to $B$ and back gives the start point, and going from $B$ to $A$ and back gives the start point too.\n",
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  "msc": [
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  "name": "Bijective iff it has a two-sided inverse",
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