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Definition df-clat 17029
Description: Define the class of all complete lattices, where every subset of the base set has an LUB and a GLB. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Assertion
Ref Expression
df-clat CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}

Detailed syntax breakdown of Definition df-clat
StepHypRef Expression
1 ccla 17028 . 2 class CLat
2 vp . . . . . . . 8 setvar 𝑝
32cv 1479 . . . . . . 7 class 𝑝
4 club 16863 . . . . . . 7 class lub
53, 4cfv 5847 . . . . . 6 class (lub‘𝑝)
65cdm 5074 . . . . 5 class dom (lub‘𝑝)
7 cbs 15781 . . . . . . 7 class Base
83, 7cfv 5847 . . . . . 6 class (Base‘𝑝)
98cpw 4130 . . . . 5 class 𝒫 (Base‘𝑝)
106, 9wceq 1480 . . . 4 wff dom (lub‘𝑝) = 𝒫 (Base‘𝑝)
11 cglb 16864 . . . . . . 7 class glb
123, 11cfv 5847 . . . . . 6 class (glb‘𝑝)
1312cdm 5074 . . . . 5 class dom (glb‘𝑝)
1413, 9wceq 1480 . . . 4 wff dom (glb‘𝑝) = 𝒫 (Base‘𝑝)
1510, 14wa 384 . . 3 wff (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))
16 cpo 16861 . . 3 class Poset
1715, 2, 16crab 2911 . 2 class {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
181, 17wceq 1480 1 wff CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
Colors of variables: wff setvar class
This definition is referenced by:  isclat  17030
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