Corollary 2

Tag 03KU, permanent. Cite it as [Bourbaki, Tag 03KU].

Theory of Sets, chapter III, § 3. Equipotent sets. Cardinals

Let $(\mathfrak{a}_\iota)_{\iota \in \mathbf{I}}$ be a family of cardinals and let $\mathfrak{b}$ be a cardinal. Then

\left( \mathop{\mathbf{P}}_{\iota \in \mathbf{I}} \mathfrak{a}_\iota \right)^{\mathfrak{b}} = \mathop{\mathbf{P}}_{\iota \in \mathbf{I}} \mathfrak{a}_\iota^{\mathfrak{b}}.

Let $\mathfrak{a}_{\iota\beta} = \mathfrak{a}_\iota$ for each pair $(\iota, \beta) \in \mathbf{I} \times \mathfrak{b}$ . Then, by associativity of the product, we have

\left( \mathop{\mathbf{P}}_{\iota \in \mathbf{I}} \mathfrak{a}_\iota \right)^{\mathfrak{b}} = \mathop{\mathbf{P}}_{\beta \in \mathfrak{b}} \left( \mathop{\mathbf{P}}_{\iota \in \mathbf{I}} \mathfrak{a}_{\iota\beta} \right) = \mathop{\mathbf{P}}_{\iota \in \mathbf{I}} \left( \mathop{\mathbf{P}}_{\beta \in \mathfrak{b}} \mathfrak{a}_{\iota\beta} \right) = \mathop{\mathbf{P}}_{\iota \in \mathbf{I}} \mathfrak{a}_\iota^{\mathfrak{b}}.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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