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Theorem glbeu 16977
Description: Unique existence proper of a member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
glbval.b 𝐵 = (Base‘𝐾)
glbval.l = (le‘𝐾)
glbval.g 𝐺 = (glb‘𝐾)
glbval.p (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
glbva.k (𝜑𝐾𝑉)
glbval.s (𝜑𝑆 ∈ dom 𝐺)
Assertion
Ref Expression
glbeu (𝜑 → ∃!𝑥𝐵 𝜓)
Distinct variable groups:   𝑥,𝑧,𝐵   𝑥,𝑦,𝐾,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐵(𝑦)   𝐺(𝑥,𝑦,𝑧)   (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem glbeu
StepHypRef Expression
1 glbval.s . . 3 (𝜑𝑆 ∈ dom 𝐺)
2 glbval.b . . . 4 𝐵 = (Base‘𝐾)
3 glbval.l . . . 4 = (le‘𝐾)
4 glbval.g . . . 4 𝐺 = (glb‘𝐾)
5 glbval.p . . . 4 (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
6 glbva.k . . . 4 (𝜑𝐾𝑉)
72, 3, 4, 5, 6glbeldm 16975 . . 3 (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 𝜓)))
81, 7mpbid 222 . 2 (𝜑 → (𝑆𝐵 ∧ ∃!𝑥𝐵 𝜓))
98simprd 479 1 (𝜑 → ∃!𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  ∃!wreu 2911  wss 3567   class class class wbr 4644  dom cdm 5104  cfv 5876  Basecbs 15838  lecple 15929  glbcglb 16924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-glb 16956
This theorem is referenced by:  glbval  16978  glbcl  16979  glbprop  16980  meeteu  17005  isglbd  17098
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