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Theorem glbcl 17606
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 2824 . . 3 (le‘𝐾) = (le‘𝐾)
3 glbc.g . . 3 𝐺 = (glb‘𝐾)
4 biid 264 . . 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 17603 . . 3 (𝜑𝑆𝐵)
81, 2, 3, 4, 5, 7glbval 17605 . 2 (𝜑 → (𝐺𝑆) = (𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))))
91, 2, 3, 4, 5, 6glbeu 17604 . . 3 (𝜑 → ∃!𝑥𝐵 (∀𝑦𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
10 riotacl 7121 . . 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 2916 1 (𝜑 → (𝐺𝑆) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  ∃!wreu 3135   class class class wbr 5053  dom cdm 5543  cfv 6344  crio 7103  Basecbs 16481  lecple 16570  glbcglb 17551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-glb 17583
This theorem is referenced by:  glbprop  17607  meetcl  17628  clatlem  17719  op0cl  36392  atl0cl  36511
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