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Theorem lubelss 18353
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 𝐡 = (Baseβ€˜πΎ)
lubs.l ≀ = (leβ€˜πΎ)
lubs.u π‘ˆ = (lubβ€˜πΎ)
lubs.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
lubs.s (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)
Assertion
Ref Expression
lubelss (πœ‘ β†’ 𝑆 βŠ† 𝐡)

Proof of Theorem lubelss
Dummy variables π‘₯ 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubs.s . . 3 (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)
2 lubs.b . . . 4 𝐡 = (Baseβ€˜πΎ)
3 lubs.l . . . 4 ≀ = (leβ€˜πΎ)
4 lubs.u . . . 4 π‘ˆ = (lubβ€˜πΎ)
5 biid 260 . . . 4 ((βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)) ↔ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))
6 lubs.k . . . 4 (πœ‘ β†’ 𝐾 ∈ 𝑉)
72, 3, 4, 5, 6lubeldm 18352 . . 3 (πœ‘ β†’ (𝑆 ∈ dom π‘ˆ ↔ (𝑆 βŠ† 𝐡 ∧ βˆƒ!π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))))
81, 7mpbid 231 . 2 (πœ‘ β†’ (𝑆 βŠ† 𝐡 ∧ βˆƒ!π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧))))
98simpld 493 1 (πœ‘ β†’ 𝑆 βŠ† 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒ!wreu 3372   βŠ† wss 3949   class class class wbr 5152  dom cdm 5682  β€˜cfv 6553  Basecbs 17187  lecple 17247  lubclub 18308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-lub 18345
This theorem is referenced by:  lubcl  18356  lubprop  18357  joinfval  18372  joindmss  18378  lubsscl  48057
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