Theorem ssiun

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	ssiun	$$⊢\exists x\in AC\subseteq B\to C\subseteq \bigcup _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢C\subseteq B\to \left(y\in C\to y\in B\right)$$
2	1	reximi	$$⊢\exists x\in AC\subseteq B\to \exists x\in A\left(y\in C\to y\in B\right)$$
3		r19.37v	$$⊢\exists x\in A\left(y\in C\to y\in B\right)\to \left(y\in C\to \exists x\in Ay\in B\right)$$
4	2 3	syl	$$⊢\exists x\in AC\subseteq B\to \left(y\in C\to \exists x\in Ay\in B\right)$$
5		eliun	$$⊢y\in \bigcup _{x\in A}B\leftrightarrow \exists x\in Ay\in B$$
6	4 5	syl6ibr	$$⊢\exists x\in AC\subseteq B\to \left(y\in C\to y\in \bigcup _{x\in A}B\right)$$
7	6	ssrdv	$$⊢\exists x\in AC\subseteq B\to C\subseteq \bigcup _{x\in A}B$$