To find out what portion of the total amount of work Dan has completed, we can follow these steps:
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Determine Dan's share of the total work: Dan's share of the work is given as \( \frac{3}{7} \) of the team's total work.
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Calculate the fraction of the total work Dan has completed: Dan has completed \( \frac{2}{5} \) of his share of work. Therefore, the portion of the total work he completed can be calculated by multiplying his share by the fraction he has completed:
\[ \text{Portion of total work completed by Dan} = \frac{2}{5} \times \frac{3}{7} \]
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Perform the multiplication: To multiply these fractions, multiply the numerators and then the denominators:
\[ = \frac{2 \times 3}{5 \times 7} = \frac{6}{35} \]
Thus, the portion of the total amount of work that Dan has completed is \( \frac{6}{35} \).
So, the final answer is:
\[ \boxed{\frac{6}{35}} \]