Theorem iinss

Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	iinss	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

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		ssel	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
4	3	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
5		r19.36v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
6	4 5	syl	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
7	2 6	syl5bi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐶 ) )
8	7	ssrdv	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )