Definition df-glb 18074

Description: Define the greatest lower bound (GLB) of a set of (poset) elements. The domain is restricted to exclude sets 𝑠 for which the GLB doesn't exist uniquely. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)

Assertion

Ref	Expression
df-glb	⊢ glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}))

Distinct variable group:   𝑠,𝑝,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-glb

Step	Hyp	Ref	Expression
1		cglb 18037	. 2 class glb
2		vp	. . 3 setvar 𝑝
3		cvv 3433	. . 3 class V
4		vs	. . . . 5 setvar 𝑠
5	2	cv 1538	. . . . . . 7 class 𝑝
6		cbs 16921	. . . . . . 7 class Base
7	5, 6	cfv 6437	. . . . . 6 class (Base‘𝑝)
8	7	cpw 4534	. . . . 5 class 𝒫 (Base‘𝑝)
9		vx	. . . . . . . . . 10 setvar 𝑥
10	9	cv 1538	. . . . . . . . 9 class 𝑥
11		vy	. . . . . . . . . 10 setvar 𝑦
12	11	cv 1538	. . . . . . . . 9 class 𝑦
13		cple 16978	. . . . . . . . . 10 class le
14	5, 13	cfv 6437	. . . . . . . . 9 class (le‘𝑝)
15	10, 12, 14	wbr 5075	. . . . . . . 8 wff 𝑥(le‘𝑝)𝑦
16	4	cv 1538	. . . . . . . 8 class 𝑠
17	15, 11, 16	wral 3065	. . . . . . 7 wff ∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦
18		vz	. . . . . . . . . . . 12 setvar 𝑧
19	18	cv 1538	. . . . . . . . . . 11 class 𝑧
20	19, 12, 14	wbr 5075	. . . . . . . . . 10 wff 𝑧(le‘𝑝)𝑦
21	20, 11, 16	wral 3065	. . . . . . . . 9 wff ∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦
22	19, 10, 14	wbr 5075	. . . . . . . . 9 wff 𝑧(le‘𝑝)𝑥
23	21, 22	wi 4	. . . . . . . 8 wff (∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)
24	23, 18, 7	wral 3065	. . . . . . 7 wff ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)
25	17, 24	wa 396	. . . . . 6 wff (∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))
26	25, 9, 7	crio 7240	. . . . 5 class (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))
27	4, 8, 26	cmpt 5158	. . . 4 class (𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))))
28	25, 9, 7	wreu 3067	. . . . 5 wff ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))
29	28, 4	cab 2716	. . . 4 class {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}
30	27, 29	cres 5592	. . 3 class ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))})
31	2, 3, 30	cmpt 5158	. 2 class (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}))
32	1, 31	wceq 1539	1 wff glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}))

This definition is referenced by:  glbfval  18090