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Theorem glbelss 16976
 Description: A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
glbs.b 𝐵 = (Base‘𝐾)
glbs.l = (le‘𝐾)
glbs.g 𝐺 = (glb‘𝐾)
glbs.k (𝜑𝐾𝑉)
glbs.s (𝜑𝑆 ∈ dom 𝐺)
Assertion
Ref Expression
glbelss (𝜑𝑆𝐵)

Proof of Theorem glbelss
Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 glbs.s . . 3 (𝜑𝑆 ∈ dom 𝐺)
2 glbs.b . . . 4 𝐵 = (Base‘𝐾)
3 glbs.l . . . 4 = (le‘𝐾)
4 glbs.g . . . 4 𝐺 = (glb‘𝐾)
5 biid 251 . . . 4 ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
6 glbs.k . . . 4 (𝜑𝐾𝑉)
72, 3, 4, 5, 6glbeldm 16975 . . 3 (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))))
81, 7mpbid 222 . 2 (𝜑 → (𝑆𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))))
98simpld 475 1 (𝜑𝑆𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ 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:  glbcl  16979  glbprop  16980  meetfval  16996  meetdmss  17002
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