Theorem iinssd

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

	Ref	Expression
Hypotheses	iinssd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	iinssd.2	⊢ ( 𝑥 = 𝑋 → 𝐵 = 𝐷 )
	iinssd.3	⊢ ( 𝜑 → 𝐷 ⊆ 𝐶 )
Assertion	iinssd	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iinssd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
2		iinssd.2	⊢ ( 𝑥 = 𝑋 → 𝐵 = 𝐷 )
3		iinssd.3	⊢ ( 𝜑 → 𝐷 ⊆ 𝐶 )
4	2	sseq1d	⊢ ( 𝑥 = 𝑋 → ( 𝐵 ⊆ 𝐶 ↔ 𝐷 ⊆ 𝐶 ) )
5	4	rspcev	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶 ) → ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
6	1 3 5	syl2anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
7		iinss	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
8	6 7	syl	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )