Theorem eliunid

Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Assertion	eliunid	$$⊢\left(x\in A\wedge C\in B\right)\to C\in \bigcup _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		rspe	$$⊢\left(x\in A\wedge C\in B\right)\to \exists x\in AC\in B$$
2		eliun	$$⊢C\in \bigcup _{x\in A}B\leftrightarrow \exists x\in AC\in B$$
3	1 2	sylibr	$$⊢\left(x\in A\wedge C\in B\right)\to C\in \bigcup _{x\in A}B$$