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Theorem clatlubcl2 17029
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 6160 . . . . . 6 (Base‘𝐾) ∈ V
31, 2eqeltri 2700 . . . . 5 𝐵 ∈ V
43elpw2 4793 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
54biimpri 218 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
65adantl 482 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
7 clatlubcl.u . . . . 5 𝑈 = (lub‘𝐾)
8 eqid 2626 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
91, 7, 8isclat 17025 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)))
10 simprl 793 . . . 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 481 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝑈 = 𝒫 𝐵)
136, 12eleqtrrd 2707 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  wss 3560  𝒫 cpw 4135  dom cdm 5079  cfv 5850  Basecbs 15776  Posetcpo 16856  lubclub 16858  glbcglb 16859  CLatccla 17023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-dm 5089  df-iota 5813  df-fv 5858  df-clat 17024
This theorem is referenced by:  lublem  17034
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