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  "body": "\n## Proof\n\nWe show that each element of the left side is an element of the right side.\n\n1. Let $f \\colon A \\to B$ be a function and let $S, T \\subseteq A$.\n2. Let $y \\in f(S \\cap T)$. By the definition of the [[image|0Q3F20XB0A]], there is $s \\in S \\cap T$ with $f(s) = y$.\n3. By the definition of [[intersection|0HBRW4E6G1]], $s \\in S$ and $s \\in T$.\n4. Since $s \\in S$ and $f(s) = y$, the definition of the image gives $y \\in f(S)$.\n5. Since $s \\in T$ and $f(s) = y$, the definition of the image gives $y \\in f(T)$.\n6. By steps 4 and 5, $y \\in f(S) \\cap f(T)$.\n7. Step 2 took an arbitrary element $y$ of $f(S \\cap T)$. So $f(S \\cap T)$ is a [[subset|0FE8Q5S42Y]] of $f(S) \\cap f(T)$.\n\n## Idea\n\nA point of $S \\cap T$ that maps to $y$ is a witness in $S$ and a witness in $T$ at the same time. The reverse inclusion can fail, because $y$ may come from one point of $S$ and a different point of $T$.\n",
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