Theorem uniixp

Description: The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006) (Revised by Mario Carneiro, 24-Jun-2015)

	Ref	Expression
Assertion	uniixp	$$⊢\bigcup \underset{x\in A}{⨉}B\subseteq A\times \bigcup _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		ixpf	$$⊢f\in \underset{x\in A}{⨉}B\to f:A⟶\bigcup _{x\in A}B$$
2		fssxp	$$⊢f:A⟶\bigcup _{x\in A}B\to f\subseteq A\times \bigcup _{x\in A}B$$
3	1 2	syl	$$⊢f\in \underset{x\in A}{⨉}B\to f\subseteq A\times \bigcup _{x\in A}B$$
4		velpw	$$⊢f\in 𝒫\left(A\times \bigcup _{x\in A}B\right)\leftrightarrow f\subseteq A\times \bigcup _{x\in A}B$$
5	3 4	sylibr	$$⊢f\in \underset{x\in A}{⨉}B\to f\in 𝒫\left(A\times \bigcup _{x\in A}B\right)$$
6	5	ssriv	$$⊢\underset{x\in A}{⨉}B\subseteq 𝒫\left(A\times \bigcup _{x\in A}B\right)$$
7		sspwuni	$$⊢\underset{x\in A}{⨉}B\subseteq 𝒫\left(A\times \bigcup _{x\in A}B\right)\leftrightarrow \bigcup \underset{x\in A}{⨉}B\subseteq A\times \bigcup _{x\in A}B$$
8	6 7	mpbi	$$⊢\bigcup \underset{x\in A}{⨉}B\subseteq A\times \bigcup _{x\in A}B$$