Theorem iinssf

Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	iinssf.1	⊢ Ⅎ 𝑥 𝐶
Assertion	iinssf	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iinssf.1	⊢ Ⅎ 𝑥 𝐶
2		eliin	⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
3	2	elv	⊢ ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
4		ssel	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
5	4	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
6	1	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐶
7	6	r19.36vf	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
8	5 7	syl	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
9	3 8	syl5bi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐶 ) )
10	9	ssrdv	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )