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Theorem glbfun 18410
Description: The GLB is a function. (Contributed by NM, 9-Sep-2018.)
Hypothesis
Ref Expression
glbfun.g 𝐺 = (glb‘𝐾)
Assertion
Ref Expression
glbfun Fun 𝐺

Proof of Theorem glbfun
Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 6604 . . . 4 Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))))
2 funres 6608 . . . 4 (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))}))
31, 2ax-mp 5 . . 3 Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))})
4 eqid 2737 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2737 . . . . 5 (le‘𝐾) = (le‘𝐾)
6 glbfun.g . . . . 5 𝐺 = (glb‘𝐾)
7 biid 261 . . . . 5 ((∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
8 id 22 . . . . 5 (𝐾 ∈ V → 𝐾 ∈ V)
94, 5, 6, 7, 8glbfval 18408 . . . 4 (𝐾 ∈ V → 𝐺 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))}))
109funeqd 6588 . . 3 (𝐾 ∈ V → (Fun 𝐺 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))})))
113, 10mpbiri 258 . 2 (𝐾 ∈ V → Fun 𝐺)
12 fun0 6631 . . 3 Fun ∅
13 fvprc 6898 . . . . 5 𝐾 ∈ V → (glb‘𝐾) = ∅)
146, 13eqtrid 2789 . . . 4 𝐾 ∈ V → 𝐺 = ∅)
1514funeqd 6588 . . 3 𝐾 ∈ V → (Fun 𝐺 ↔ Fun ∅))
1612, 15mpbiri 258 . 2 𝐾 ∈ V → Fun 𝐺)
1711, 16pm2.61i 182 1 Fun 𝐺
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  ∃!wreu 3378  Vcvv 3480  c0 4333  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  cres 5687  Fun wfun 6555  cfv 6561  crio 7387  Basecbs 17247  lecple 17304  glbcglb 18356
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  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-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-glb 18392
This theorem is referenced by:  meetfval  18432  meetfval2  18433
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