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Theorem lubsscl 48857
Description: If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
Hypotheses
Ref Expression
lubsscl.k (𝜑𝐾 ∈ Poset)
lubsscl.t (𝜑𝑇𝑆)
lubsscl.u 𝑈 = (lub‘𝐾)
lubsscl.s (𝜑𝑆 ∈ dom 𝑈)
lubsscl.x (𝜑 → (𝑈𝑆) ∈ 𝑇)
Assertion
Ref Expression
lubsscl (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈𝑇) = (𝑈𝑆)))

Proof of Theorem lubsscl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubsscl.t . . . 4 (𝜑𝑇𝑆)
2 eqid 2737 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2737 . . . . 5 (le‘𝐾) = (le‘𝐾)
4 lubsscl.u . . . . 5 𝑈 = (lub‘𝐾)
5 lubsscl.k . . . . 5 (𝜑𝐾 ∈ Poset)
6 lubsscl.s . . . . 5 (𝜑𝑆 ∈ dom 𝑈)
72, 3, 4, 5, 6lubelss 18399 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐾))
81, 7sstrd 3994 . . 3 (𝜑𝑇 ⊆ (Base‘𝐾))
9 lubsscl.x . . . . 5 (𝜑 → (𝑈𝑆) ∈ 𝑇)
108, 9sseldd 3984 . . . 4 (𝜑 → (𝑈𝑆) ∈ (Base‘𝐾))
115adantr 480 . . . . . 6 ((𝜑𝑦𝑇) → 𝐾 ∈ Poset)
126adantr 480 . . . . . 6 ((𝜑𝑦𝑇) → 𝑆 ∈ dom 𝑈)
131sselda 3983 . . . . . 6 ((𝜑𝑦𝑇) → 𝑦𝑆)
142, 3, 4, 11, 12, 13luble 18404 . . . . 5 ((𝜑𝑦𝑇) → 𝑦(le‘𝐾)(𝑈𝑆))
1514ralrimiva 3146 . . . 4 (𝜑 → ∀𝑦𝑇 𝑦(le‘𝐾)(𝑈𝑆))
16 breq1 5146 . . . . . . 7 (𝑦 = (𝑈𝑆) → (𝑦(le‘𝐾)𝑧 ↔ (𝑈𝑆)(le‘𝐾)𝑧))
17 simp3 1139 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦𝑇 𝑦(le‘𝐾)𝑧) → ∀𝑦𝑇 𝑦(le‘𝐾)𝑧)
1893ad2ant1 1134 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦𝑇 𝑦(le‘𝐾)𝑧) → (𝑈𝑆) ∈ 𝑇)
1916, 17, 18rspcdva 3623 . . . . . 6 ((𝜑𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦𝑇 𝑦(le‘𝐾)𝑧) → (𝑈𝑆)(le‘𝐾)𝑧)
20193expia 1122 . . . . 5 ((𝜑𝑧 ∈ (Base‘𝐾)) → (∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧))
2120ralrimiva 3146 . . . 4 (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧))
22 breq2 5147 . . . . . . 7 (𝑥 = (𝑈𝑆) → (𝑦(le‘𝐾)𝑥𝑦(le‘𝐾)(𝑈𝑆)))
2322ralbidv 3178 . . . . . 6 (𝑥 = (𝑈𝑆) → (∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ↔ ∀𝑦𝑇 𝑦(le‘𝐾)(𝑈𝑆)))
24 breq1 5146 . . . . . . . 8 (𝑥 = (𝑈𝑆) → (𝑥(le‘𝐾)𝑧 ↔ (𝑈𝑆)(le‘𝐾)𝑧))
2524imbi2d 340 . . . . . . 7 (𝑥 = (𝑈𝑆) → ((∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧) ↔ (∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧)))
2625ralbidv 3178 . . . . . 6 (𝑥 = (𝑈𝑆) → (∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧) ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧)))
2723, 26anbi12d 632 . . . . 5 (𝑥 = (𝑈𝑆) → ((∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦𝑇 𝑦(le‘𝐾)(𝑈𝑆) ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧))))
2827rspcev 3622 . . . 4 (((𝑈𝑆) ∈ (Base‘𝐾) ∧ (∀𝑦𝑇 𝑦(le‘𝐾)(𝑈𝑆) ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧))) → ∃𝑥 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
2910, 15, 21, 28syl12anc 837 . . 3 (𝜑 → ∃𝑥 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
30 biid 261 . . . 4 ((∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
312, 3, 4, 30, 5lubeldm2 48853 . . 3 (𝜑 → (𝑇 ∈ dom 𝑈 ↔ (𝑇 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))))
328, 29, 31mpbir2and 713 . 2 (𝜑𝑇 ∈ dom 𝑈)
333, 2, 4, 5, 8, 10, 14, 19poslubd 18458 . 2 (𝜑 → (𝑈𝑇) = (𝑈𝑆))
3432, 33jca 511 1 (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈𝑇) = (𝑈𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   class class class wbr 5143  dom cdm 5685  cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  lubclub 18355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-proset 18340  df-poset 18359  df-lub 18391
This theorem is referenced by:  lubprlem  48859
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