MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  glbfun Structured version   Visualization version   GIF version

Theorem glbfun 17201
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 6069 . . . 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 6072 . . . 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 2771 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2771 . . . . 5 (le‘𝐾) = (le‘𝐾)
6 glbfun.g . . . . 5 𝐺 = (glb‘𝐾)
7 biid 251 . . . . 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 17199 . . . 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 6053 . . 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 248 . 2 (𝐾 ∈ V → Fun 𝐺)
12 fun0 6094 . . 3 Fun ∅
13 fvprc 6326 . . . . 5 𝐾 ∈ V → (glb‘𝐾) = ∅)
146, 13syl5eq 2817 . . . 4 𝐾 ∈ V → 𝐺 = ∅)
1514funeqd 6053 . . 3 𝐾 ∈ V → (Fun 𝐺 ↔ Fun ∅))
1612, 15mpbiri 248 . 2 𝐾 ∈ V → Fun 𝐺)
1711, 16pm2.61i 176 1 Fun 𝐺
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wral 3061  ∃!wreu 3063  Vcvv 3351  c0 4063  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  cres 5251  Fun wfun 6025  cfv 6031  crio 6753  Basecbs 16064  lecple 16156  glbcglb 17151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-glb 17183
This theorem is referenced by:  meetfval  17223  meetfval2  17224
  Copyright terms: Public domain W3C validator