Corollary

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

Theory of Sets, chapter II, § 5. Product of a family of sets

Let $(X_\iota)_{\iota \in I}$ and $(Y_\iota)_{\iota \in I}$ be two families of sets with the same index set $I \neq \emptyset$ . Then

\Big(\prod_{\iota \in I} X_\iota\Big) \cap \Big(\prod_{\iota \in I} Y_\iota\Big) = \prod_{\iota \in I} (X_\iota \cap Y_\iota),

\Big(\bigcap_{\iota \in I} X_\iota\Big) \times \Big(\bigcap_{\iota \in I} Y_\iota\Big) = \bigcap_{\iota \in I} (X_\iota \times Y_\iota).

Apply Proposition 10 to the case where K (resp. I) is a set consisting of two distinct elements.

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.

Cited by

Nothing.

Cites

Proposition 10