Theorem dfiin3

Description: Alternate definition of indexed intersection when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypothesis	dfiun3.1	$$⊢B\in \mathrm{V}$$
Assertion	dfiin3	$$⊢\bigcap _{x\in A}B=\bigcap \mathrm{ran}\left(x\in A⟼B\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfiun3.1	$$⊢B\in \mathrm{V}$$
2		dfiin3g	$$⊢\forall x\in AB\in \mathrm{V}\to \bigcap _{x\in A}B=\bigcap \mathrm{ran}\left(x\in A⟼B\right)$$
3	1	a1i	$$⊢x\in A\to B\in \mathrm{V}$$
4	2 3	mprg	$$⊢\bigcap _{x\in A}B=\bigcap \mathrm{ran}\left(x\in A⟼B\right)$$