Theorem iinglb 49620

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 3968	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐵 ⊆ 𝐶 ↔ 𝐶 ⊆ 𝐶))
4		ssidd 3960	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐶)
5	1, 3, 4	rspcedvd 3583	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		iinss 5021	. . 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 3157	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
10		ssiin 5020	. . 3 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
11	9, 10	sylibr 237	. 2 ⊢ (𝜑 → 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
12	7, 11	eqssd 3954	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  ∩ ciin 4957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3457  df-ss 3922  df-iin 4959

This theorem is referenced by:  iinfconstbas  49864