Theorem dfiin2

Description: Alternate definition of indexed intersection when B is a set. Definition 15(b) of Suppes p. 44. (Contributed by NM, 28-Jun-1998) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Hypothesis	dfiun2.1	$$⊢B\in \mathrm{V}$$
Assertion	dfiin2	$$⊢\bigcap _{x\in A}B=\bigcap \left\{y|\exists x\in Ay=B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfiun2.1	$$⊢B\in \mathrm{V}$$
2		dfiin2g	$$⊢\forall x\in AB\in \mathrm{V}\to \bigcap _{x\in A}B=\bigcap \left\{y|\exists x\in Ay=B\right\}$$
3	1	a1i	$$⊢x\in A\to B\in \mathrm{V}$$
4	2 3	mprg	$$⊢\bigcap _{x\in A}B=\bigcap \left\{y|\exists x\in Ay=B\right\}$$