Theorem iinglb 49659

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

Hypotheses

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

Assertion

Ref	Expression
iinglb	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iinglb

Step	Hyp	Ref	Expression
1		iunlub.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		iunlub.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)
3	2	sseq1d 3969	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐵 ⊆ 𝐶 ↔ 𝐶 ⊆ 𝐶))
4		ssidd 3961	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐶)
5	1, 3, 4	rspcedvd 3585	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		iinss 5023	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7	5, 6	syl 18	. 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
8		iinglb.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐵)
9	8	ralrimiva 3159	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
10		ssiin 5022	. . 3 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
11	9, 10	sylibr 237	. 2 ⊢ (𝜑 → 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
12	7, 11	eqssd 3955	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  ∩ ciin 4959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-v 3459  df-ss 3923  df-iin 4961

This theorem is used by:  iinfconstbas  49903