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Theorem clatlubcl2 17802
 Description: Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
clatlubcl.b 𝐵 = (Base‘𝐾)
clatlubcl.u 𝑈 = (lub‘𝐾)
Assertion
Ref Expression
clatlubcl2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)

Proof of Theorem clatlubcl2
StepHypRef Expression
1 clatlubcl.b . . . . . 6 𝐵 = (Base‘𝐾)
21fvexi 6677 . . . . 5 𝐵 ∈ V
32elpw2 5219 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
43biimpri 231 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
54adantl 485 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
6 clatlubcl.u . . . . 5 𝑈 = (lub‘𝐾)
7 eqid 2758 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
81, 6, 7isclat 17798 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)))
9 simprl 770 . . . 4 ((𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)) → dom 𝑈 = 𝒫 𝐵)
108, 9sylbi 220 . . 3 (𝐾 ∈ CLat → dom 𝑈 = 𝒫 𝐵)
1110adantr 484 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝑈 = 𝒫 𝐵)
125, 11eleqtrrd 2855 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  𝒫 cpw 4497  dom cdm 5528  ‘cfv 6340  Basecbs 16554  Posetcpo 17629  lubclub 17631  glbcglb 17632  CLatccla 17796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-dm 5538  df-iota 6299  df-fv 6348  df-clat 17797 This theorem is referenced by:  lublem  17807
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