Theorem iinss1

Description: Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012)

	Ref	Expression
Assertion	iinss1	⊢ ( 𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
2		eliin	⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ) )
3	2	elv	⊢ ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 )
4		eliin	⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
5	4	elv	⊢ ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 )
6	1 3 5	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ) )
7	6	ssrdv	⊢ ( 𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶 )