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I am trying to proof now that $\mathcal P(\Bbb N)$ is of the same cardinality as $\mathcal P(\Bbb N)^\Bbb N$ - the set of all functions $f:\Bbb N\to\mathcal P(\Bbb N)$. Currently I've tried to use the knowledge that: $\mathcal P(\Bbb N)\sim \{0,1\}^\Bbb N$ but it doesn't help much. Also, I've proved that: $\left(A^B\right)^C \sim A^{B\times C}$.

Any ideas how to continue? Thanks in advance.

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HINT: Note that from the last thing you mention it follows that $\left(\{0,1\}^\Bbb N\right)^\Bbb N=\{0,1\}^{\Bbb{N\times N}}$, and that $\Bbb{N\sim N\times N}$.

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