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Theorem clatlubcl2 17334
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)
2 fvex 6363 . . . . . 6 (Base‘𝐾) ∈ V
31, 2eqeltri 2835 . . . . 5 𝐵 ∈ V
43elpw2 4977 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
54biimpri 218 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
65adantl 473 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
7 clatlubcl.u . . . . 5 𝑈 = (lub‘𝐾)
8 eqid 2760 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
91, 7, 8isclat 17330 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)))
10 simprl 811 . . . 4 ((𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)) → dom 𝑈 = 𝒫 𝐵)
119, 10sylbi 207 . . 3 (𝐾 ∈ CLat → dom 𝑈 = 𝒫 𝐵)
1211adantr 472 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝑈 = 𝒫 𝐵)
136, 12eleqtrrd 2842 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302  dom cdm 5266  cfv 6049  Basecbs 16079  Posetcpo 17161  lubclub 17163  glbcglb 17164  CLatccla 17328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-clat 17329
This theorem is referenced by:  lublem  17339
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