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Theorem glbelss 18329
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 𝐡 = (Baseβ€˜πΎ)
glbs.l ≀ = (leβ€˜πΎ)
glbs.g 𝐺 = (glbβ€˜πΎ)
glbs.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
glbs.s (πœ‘ β†’ 𝑆 ∈ dom 𝐺)
Assertion
Ref Expression
glbelss (πœ‘ β†’ 𝑆 βŠ† 𝐡)

Proof of Theorem glbelss
Dummy variables π‘₯ 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 glbs.s . . 3 (πœ‘ β†’ 𝑆 ∈ dom 𝐺)
2 glbs.b . . . 4 𝐡 = (Baseβ€˜πΎ)
3 glbs.l . . . 4 ≀ = (leβ€˜πΎ)
4 glbs.g . . . 4 𝐺 = (glbβ€˜πΎ)
5 biid 261 . . . 4 ((βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))
6 glbs.k . . . 4 (πœ‘ β†’ 𝐾 ∈ 𝑉)
72, 3, 4, 5, 6glbeldm 18328 . . 3 (πœ‘ β†’ (𝑆 ∈ dom 𝐺 ↔ (𝑆 βŠ† 𝐡 ∧ βˆƒ!π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))))
81, 7mpbid 231 . 2 (πœ‘ β†’ (𝑆 βŠ† 𝐡 ∧ βˆƒ!π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯))))
98simpld 494 1 (πœ‘ β†’ 𝑆 βŠ† 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒ!wreu 3368   βŠ† wss 3943   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536  Basecbs 17150  lecple 17210  glbcglb 18272
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-glb 18309
This theorem is referenced by:  glbcl  18332  glbprop  18333  meetfval  18349  meetdmss  18355  glbsscl  47850
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