Theorem unilbeu 45887

Description: Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)

Assertion

Ref	Expression
unilbeu	⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem unilbeu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3912	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐴))
2		simpll 767	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ 𝐵)
3		simplr 769	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ⊆ 𝐴)
4	1, 2, 3	elrabd 3593	. . . . . 6 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ⊆ 𝐴})
5		sseq1 3912	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
6	5	cbvrabv 3392	. . . . . 6 ⊢ {𝑧 ∈ 𝐵 ∣ 𝑧 ⊆ 𝐴} = {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}
7	4, 6	eleqtrdi 2841	. . . . 5 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
8		elssuni 4837	. . . . 5 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝐶 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
9	7, 8	syl 17	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10		unissb 4839	. . . . . . 7 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐶)
11		sseq1 3912	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
12	11	ralrab 3596	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
13	10, 12	bitri 278	. . . . . 6 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
14	13	biimpri 231	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
15	14	adantl 485	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
16	9, 15	eqssd 3904	. . 3 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
17	16	expl 461	. 2 ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
18		unilbss 45779	. . . 4 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
19		sseq1 3912	. . . 4 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → (𝐶 ⊆ 𝐴 ↔ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴))
20	18, 19	mpbiri 261	. . 3 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝐶 ⊆ 𝐴)
21		eqimss2 3944	. . . 4 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
22	21, 13	sylib 221	. . 3 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
23	20, 22	jca 515	. 2 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → (𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)))
24	17, 23	impbid1 228	1 ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  {crab 3055   ⊆ wss 3853  ∪ cuni 4805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rab 3060  df-v 3400  df-in 3860  df-ss 3870  df-uni 4806

This theorem is referenced by:  ipoglbdm  45892  ipoglb  45893