$\bigcup_{i \in I} \mathcal{P} (A_i)$

Question

This is Velleman 3.7, Problem 4

Below is the problem, verbatim.

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Suppose $ \{ A_i \mid i \in I\}$ is a family of sets. Prove that if $\mathcal{P}(\bigcup_{i \in I} A_i) \subseteq \bigcup_{i \in I} \mathcal{P}(A_i)$, then there is some $ i \in I$ such that $\forall j \in I (A_j \subseteq A_i)$.

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My question is not on the proof itself, but on the notation.

If a concrete instance of the family were $\{ \{2\}, \{2, 3\}\} $, are the following correct?

$ \bigcup A_i = \{2, 3\} $

$\mathcal{P}(\bigcup_{i \in I} A_i) = \{ \emptyset, \{2\}, \{3\}, \{2, 3\}\}$

$ \mathcal{P}(A_i) = \{ \emptyset, \{\{2\}\}, \{\{2, 3\}\}, \{ \{2\}, \{2, 3\} \} \} $

$ \bigcup_{i \in I} \mathcal{P}(A_i) = \{ \{2\}, \{2, 3\} \} $

I want to make sure I understand the problem first before attempting the proof.

Answer 1

I got the answer from Mathematica.

Clear[A, B, F];

A = Sort[{{2}, {2, 3}}];

Union[A] = {{2}, {2, 3}}

Subsets[Union[A]] = {{}, {{2}}, {{2, 3}}, {{2}, {2, 3}}}

Subsets[A] = {{}, {{2}}, {{2, 3}}, {{2}, {2, 3}}}

Union[Subsets[A]] = {{}, {{2}}, {{2, 3}}, {{2}, {2, 3}}}