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  "body": "\n## Proof\n\nWe build the map in both directions and check that the two composites are identity functions.\n\n1. Let $A$ be a set. Let $E$ be the set of [[equivalence relations|0SGTMBEWRF]] on $A$, and let $\\Pi$ be the set of [[partitions|09CCE86A6D]] of $A$.\n2. For $\\sim$ in $E$, let $\\Phi(\\sim) = A/{\\sim} = \\set{[a] \\mid a \\in A}$ be the [[quotient set|0ZVHXDSZC7]], where $[a]$ is the [[equivalence class|082RMRSY26]] of $a$.\n3. By the theorem that [[equivalence classes form a partition|01HT2CH8BH]], $\\Phi(\\sim) \\in \\Pi$. So $\\Phi$ is a function from $E$ to $\\Pi$.\n4. Let $P \\in \\Pi$. Each $a \\in A$ lies in some block, because the union of the blocks is $A$.\n5. In step 4, $a$ lies in only one block. Indeed, if $a \\in B_1 \\cap B_2$ for blocks $B_1 \\neq B_2$, then $B_1$ and $B_2$ are not [[disjoint|0W4RPA8QH3]], against the definition of a partition.\n6. For $P \\in \\Pi$, define the [[binary relation|08EJC6Q25H]] $\\sim_P$ on $A$ by $a \\sim_P b$ when some block of $P$ contains both $a$ and $b$.\n7. The relation $\\sim_P$ is reflexive. By step 4, each $a$ lies in some block, which contains both $a$ and $a$.\n8. The relation $\\sim_P$ is symmetric, because the condition in step 6 does not depend on the order of $a$ and $b$.\n9. The relation $\\sim_P$ is transitive. Let $a \\sim_P b$ and $b \\sim_P c$, with $a, b \\in B_1$ and $b, c \\in B_2$ for blocks $B_1, B_2$. By step 5, $b$ lies in only one block, so $B_1 = B_2$. Then $a, c \\in B_1$, so $a \\sim_P c$.\n10. By steps 7 to 9, $\\sim_P \\in E$. Let $\\Psi(P) = \\sim_P$. So $\\Psi$ is a function from $\\Pi$ to $E$.\n11. Let $\\sim$ be in $E$, and write $\\approx$ for $\\Psi(\\Phi(\\sim))$. So $a \\approx b$ when some class $[c]$ contains both $a$ and $b$.\n12. If $a \\sim b$, then $a \\in [b]$. By reflexivity, $b \\in [b]$. So the class $[b]$ contains both, and $a \\approx b$.\n13. If $a \\approx b$, then $a, b \\in [c]$ for some $c$, so $a \\sim c$ and $b \\sim c$. By symmetry, $c \\sim b$. By transitivity, $a \\sim b$.\n14. By steps 12 and 13, $\\approx$ and $\\sim$ contain the same pairs. As subsets of $A \\times A$, they are equal [[sets|0SJ3RX9TXA]] by the [[axiom of extensionality|0TB16T2C97]]. So $\\Psi(\\Phi(\\sim)) = \\sim$ for each $\\sim$ in $E$.\n15. Let $P \\in \\Pi$. For $a \\in A$, let $B_a$ be the unique block of $P$ that contains $a$, given by steps 4 and 5. Write $[a]_P$ for the class of $a$ under $\\sim_P$.\n16. In step 15, $x \\sim_P a$ holds exactly when some block contains $x$ and $a$. By step 5, the only block containing $a$ is $B_a$. So $x \\sim_P a$ holds exactly when $x \\in B_a$. Hence $[a]_P = B_a$.\n17. By step 16, $\\Phi(\\Psi(P)) = \\set{[a]_P \\mid a \\in A} = \\set{B_a \\mid a \\in A}$. Each $B_a$ is a block of $P$, so $\\Phi(\\Psi(P)) \\subseteq P$.\n18. Let $B \\in P$. Blocks are nonempty, so pick $b \\in B$. By step 5, $B = B_b$, so $B \\in \\Phi(\\Psi(P))$. Hence $P \\subseteq \\Phi(\\Psi(P))$, and $\\Phi(\\Psi(P)) = P$.\n19. Two [[functions|0XHDTE5YMZ]] with the same domain and the same value at each point have the same pairs, so they are equal. By steps 14 and 18, the [[compositions|07HF9YCWPP]] satisfy $\\Psi \\circ \\Phi = \\mathrm{id}_E$ and $\\Phi \\circ \\Psi = \\mathrm{id}_\\Pi$, where $\\mathrm{id}$ is the [[identity function|0FSNQ0GNCC]].\n20. By the theorem that a function is [[bijective iff it has a two-sided inverse|01VG2A0BHW]], $\\Phi$ is a bijection from $E$ to $\\Pi$.\n21. Let $\\Phi^{-1}$ be the inverse of the bijection $\\Phi$, so $\\Phi^{-1} \\circ \\Phi = \\mathrm{id}_E$. By step 19 and the [[associativity of composition|0J6AKYZQT8]], $\\Phi^{-1} = \\Phi^{-1} \\circ (\\Phi \\circ \\Psi) = (\\Phi^{-1} \\circ \\Phi) \\circ \\Psi = \\Psi$. So the inverse sends each partition $P$ to $\\sim_P$.\n\n## Idea\n\nAn equivalence relation is recovered from its classes: two elements are related exactly when they share a class. A partition is recovered from its relation: the class of an element is the block that contains it. So the two constructions undo each other, and that makes each one a bijection.\n",
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