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Theorem glbcl 16938
Description: The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
glbc.b 𝐵 = (Base‘𝐾)
glbc.g 𝐺 = (glb‘𝐾)
glbc.k (𝜑𝐾𝑉)
glbc.s (𝜑𝑆 ∈ dom 𝐺)
Assertion
Ref Expression
glbcl (𝜑 → (𝐺𝑆) ∈ 𝐵)

Proof of Theorem glbcl
Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 glbc.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
3 glbc.g . . 3 𝐺 = (glb‘𝐾)
4 biid 251 . . 3 ((∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
5 glbc.k . . 3 (𝜑𝐾𝑉)
6 glbc.s . . . 4 (𝜑𝑆 ∈ dom 𝐺)
71, 2, 3, 5, 6glbelss 16935 . . 3 (𝜑𝑆𝐵)
81, 2, 3, 4, 5, 7glbval 16937 . 2 (𝜑 → (𝐺𝑆) = (𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))))
91, 2, 3, 4, 5, 6glbeu 16936 . . 3 (𝜑 → ∃!𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
10 riotacl 6590 . . 3 (∃!𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)) → (𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))) ∈ 𝐵)
119, 10syl 17 . 2 (𝜑 → (𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))) ∈ 𝐵)
128, 11eqeltrd 2698 1 (𝜑 → (𝐺𝑆) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  ∃!wreu 2910   class class class wbr 4623  dom cdm 5084  cfv 5857  crio 6575  Basecbs 15800  lecple 15888  glbcglb 16883
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-glb 16915
This theorem is referenced by:  glbprop  16939  meetcl  16960  clatlem  17051  op0cl  33990  atl0cl  34109
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