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Theorem clatlubcl2 18515
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 6910 . . . . 5 𝐵 ∈ V
32elpw2 5348 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
43biimpri 227 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
54adantl 480 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
6 clatlubcl.u . . . . 5 𝑈 = (lub‘𝐾)
7 eqid 2725 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
81, 6, 7isclat 18511 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)))
9 simprl 769 . . . 4 ((𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)) → dom 𝑈 = 𝒫 𝐵)
108, 9sylbi 216 . . 3 (𝐾 ∈ CLat → dom 𝑈 = 𝒫 𝐵)
1110adantr 479 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝑈 = 𝒫 𝐵)
125, 11eleqtrrd 2828 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3944  𝒫 cpw 4604  dom cdm 5678  cfv 6549  Basecbs 17199  Posetcpo 18318  lubclub 18320  glbcglb 18321  CLatccla 18509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-dm 5688  df-iota 6501  df-fv 6557  df-clat 18510
This theorem is referenced by:  lublem  18521
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