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Theorem isclat 18434
Description: The predicate "is a complete lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
isclat.b 𝐵 = (Base‘𝐾)
isclat.u 𝑈 = (lub‘𝐾)
isclat.g 𝐺 = (glb‘𝐾)
Assertion
Ref Expression
isclat (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Proof of Theorem isclat
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6877 . . . . . 6 (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2 isclat.u . . . . . 6 𝑈 = (lub‘𝐾)
31, 2eqtr4di 2789 . . . . 5 (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
43dmeqd 5896 . . . 4 (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5 fveq2 6877 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 isclat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2789 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87pweqd 4612 . . . 4 (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
94, 8eqeq12d 2747 . . 3 (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10 fveq2 6877 . . . . . 6 (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11 isclat.g . . . . . 6 𝐺 = (glb‘𝐾)
1210, 11eqtr4di 2789 . . . . 5 (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
1312dmeqd 5896 . . . 4 (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
1413, 8eqeq12d 2747 . . 3 (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
159, 14anbi12d 631 . 2 (𝑙 = 𝐾 → ((dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙)) ↔ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
16 df-clat 18433 . 2 CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
1715, 16elrab2 3681 1 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  𝒫 cpw 4595  dom cdm 5668  cfv 6531  Basecbs 17125  Posetcpo 18241  lubclub 18243  glbcglb 18244  CLatccla 18432
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5141  df-dm 5678  df-iota 6483  df-fv 6539  df-clat 18433
This theorem is referenced by:  clatpos  18435  clatlem  18436  clatlubcl2  18438  clatglbcl2  18440  oduclatb  18441  clatl  18442  xrsclat  32049  isclatd  47244
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