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Theorem lubfun 16912
Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)
Hypothesis
Ref Expression
lubfun.u 𝑈 = (lub‘𝐾)
Assertion
Ref Expression
lubfun Fun 𝑈

Proof of Theorem lubfun
Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 5889 . . . 4 Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))))
2 funres 5892 . . . 4 (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))}))
31, 2ax-mp 5 . . 3 Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})
4 eqid 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
6 lubfun.u . . . . 5 𝑈 = (lub‘𝐾)
7 biid 251 . . . . 5 ((∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
8 id 22 . . . . 5 (𝐾 ∈ V → 𝐾 ∈ V)
94, 5, 6, 7, 8lubfval 16910 . . . 4 (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))}))
109funeqd 5874 . . 3 (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})))
113, 10mpbiri 248 . 2 (𝐾 ∈ V → Fun 𝑈)
12 fun0 5917 . . 3 Fun ∅
13 fvprc 6147 . . . . 5 𝐾 ∈ V → (lub‘𝐾) = ∅)
146, 13syl5eq 2667 . . . 4 𝐾 ∈ V → 𝑈 = ∅)
1514funeqd 5874 . . 3 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
1612, 15mpbiri 248 . 2 𝐾 ∈ V → Fun 𝑈)
1711, 16pm2.61i 176 1 Fun 𝑈
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  ∃!wreu 2909  Vcvv 3189  c0 3896  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  cres 5081  Fun wfun 5846  cfv 5852  crio 6570  Basecbs 15792  lecple 15880  lubclub 16874
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-lub 16906
This theorem is referenced by:  joinfval  16933  joinfval2  16934
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