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  "body": "\n## Proof\n\nWe compose bijections with the counting bijections in the definition of [[cardinality|05099E1NMJ]].\n\n1. Let $n = |A|$. By definition, there is a [[bijection|0SNQEF7Z53]] $h : \\set{1, \\dots, n} \\to A$.\n2. Let $f : A \\to B$ be a bijection. A composite of bijections is a bijection, since [[injections compose|01FPXMTBX1]] and [[surjections compose|07EH4R3RVQ]].\n3. So $f \\circ h : \\set{1, \\dots, n} \\to B$ is a bijection. Thus $B$ has $n$ elements.\n4. The number of elements of a finite set is unique, so $|B| = n = |A|$.\n5. Conversely, let $|A| = |B| = n$. There are bijections $g : \\set{1, \\dots, n} \\to A$ and $k : \\set{1, \\dots, n} \\to B$.\n6. A bijection [[has a two-sided inverse|01VG2A0BHW]], so $g^{-1} : A \\to \\set{1, \\dots, n}$ exists and is a bijection.\n7. By step 2, $k \\circ g^{-1} : A \\to B$ is a bijection. This proves the converse.\n\n## Idea\n\nBoth directions pass through the standard set $\\set{1, \\dots, n}$.\nA bijection with another set carries the numbering of one set over to the other.\n",
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