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Theorem lubelss 16805
Description: A member of the domain of the least upper bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
lubs.b 𝐵 = (Base‘𝐾)
lubs.l = (le‘𝐾)
lubs.u 𝑈 = (lub‘𝐾)
lubs.k (𝜑𝐾𝑉)
lubs.s (𝜑𝑆 ∈ dom 𝑈)
Assertion
Ref Expression
lubelss (𝜑𝑆𝐵)

Proof of Theorem lubelss
Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubs.s . . 3 (𝜑𝑆 ∈ dom 𝑈)
2 lubs.b . . . 4 𝐵 = (Base‘𝐾)
3 lubs.l . . . 4 = (le‘𝐾)
4 lubs.u . . . 4 𝑈 = (lub‘𝐾)
5 biid 250 . . . 4 ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
6 lubs.k . . . 4 (𝜑𝐾𝑉)
72, 3, 4, 5, 6lubeldm 16804 . . 3 (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))))
81, 7mpbid 221 . 2 (𝜑 → (𝑆𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))))
98simpld 474 1 (𝜑𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  ∃!wreu 2898  wss 3540   class class class wbr 4583  dom cdm 5038  cfv 5804  Basecbs 15695  lecple 15775  lubclub 16765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-lub 16797
This theorem is referenced by:  lubcl  16808  lubprop  16809  joinfval  16824  joindmss  16830
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