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Theorem glbelss 17601
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 264 . . . 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 17600 . . 3 (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))))
81, 7mpbid 235 . 2 (𝜑 → (𝑆𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))))
98simpld 498 1 (𝜑𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  ∃!wreu 3135  wss 3919   class class class wbr 5052  dom cdm 5542  cfv 6343  Basecbs 16479  lecple 16568  glbcglb 17549
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
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 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-glb 17581
This theorem is referenced by:  glbcl  17604  glbprop  17605  meetfval  17621  meetdmss  17627
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