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Theorem isclat 18545
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 6906 . . . . . 6 (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2 isclat.u . . . . . 6 𝑈 = (lub‘𝐾)
31, 2eqtr4di 2795 . . . . 5 (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
43dmeqd 5916 . . . 4 (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5 fveq2 6906 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 isclat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2795 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87pweqd 4617 . . . 4 (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
94, 8eqeq12d 2753 . . 3 (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10 fveq2 6906 . . . . . 6 (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11 isclat.g . . . . . 6 𝐺 = (glb‘𝐾)
1210, 11eqtr4di 2795 . . . . 5 (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
1312dmeqd 5916 . . . 4 (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
1413, 8eqeq12d 2753 . . 3 (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
159, 14anbi12d 632 . 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 18544 . 2 CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
1715, 16elrab2 3695 1 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  𝒫 cpw 4600  dom cdm 5685  cfv 6561  Basecbs 17247  Posetcpo 18353  lubclub 18355  glbcglb 18356  CLatccla 18543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-clat 18544
This theorem is referenced by:  clatpos  18546  clatlem  18547  clatlubcl2  18549  clatglbcl2  18551  oduclatb  18552  clatl  18553  xrsclat  33013  isclatd  48872
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