Theorem iinconst

Description: Indexed intersection of a constant class, i.e. where B does not depend on x . (Contributed by Mario Carneiro, 6-Feb-2015)

	Ref	Expression
Assertion	iinconst	⊢ ( 𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eliin	⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
2	1	elv	⊢ ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
3		r19.3rzv	⊢ ( 𝐴 ≠ ∅ → ( 𝑦 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
4	2 3	bitr4id	⊢ ( 𝐴 ≠ ∅ → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
5	4	eqrdv	⊢ ( 𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵 )