Theorem iinglb

Description: The indexed intersection is the the greatest lower bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025)

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

Proof

Step	Hyp	Ref	Expression
1		iunlub.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
2		iunlub.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝐵 = 𝐶 )
3		iinglb.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ⊆ 𝐵 )
4	2	sseq1d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝐵 ⊆ 𝐶 ↔ 𝐶 ⊆ 𝐶 ) )
5		ssidd	⊢ ( 𝜑 → 𝐶 ⊆ 𝐶 )
6	1 4 5	rspcedvd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
7		iinss	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
8	6 7	syl	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
9	3	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 )
10		ssiin	⊢ ( 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 )
11	9 10	sylibr	⊢ ( 𝜑 → 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 )
12	8 11	eqssd	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )