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Theorem lubsscl 48757
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 2735 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2735 . . . . 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 18412 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐾))
81, 7sstrd 4006 . . 3 (𝜑𝑇 ⊆ (Base‘𝐾))
9 lubsscl.x . . . . 5 (𝜑 → (𝑈𝑆) ∈ 𝑇)
108, 9sseldd 3996 . . . 4 (𝜑 → (𝑈𝑆) ∈ (Base‘𝐾))
115adantr 480 . . . . . 6 ((𝜑𝑦𝑇) → 𝐾 ∈ Poset)
126adantr 480 . . . . . 6 ((𝜑𝑦𝑇) → 𝑆 ∈ dom 𝑈)
131sselda 3995 . . . . . 6 ((𝜑𝑦𝑇) → 𝑦𝑆)
142, 3, 4, 11, 12, 13luble 18417 . . . . 5 ((𝜑𝑦𝑇) → 𝑦(le‘𝐾)(𝑈𝑆))
1514ralrimiva 3144 . . . 4 (𝜑 → ∀𝑦𝑇 𝑦(le‘𝐾)(𝑈𝑆))
16 breq1 5151 . . . . . . 7 (𝑦 = (𝑈𝑆) → (𝑦(le‘𝐾)𝑧 ↔ (𝑈𝑆)(le‘𝐾)𝑧))
17 simp3 1137 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦𝑇 𝑦(le‘𝐾)𝑧) → ∀𝑦𝑇 𝑦(le‘𝐾)𝑧)
1893ad2ant1 1132 . . . . . . 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 1120 . . . . 5 ((𝜑𝑧 ∈ (Base‘𝐾)) → (∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧))
2120ralrimiva 3144 . . . 4 (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑇 𝑦(le‘𝐾)𝑧 → (𝑈𝑆)(le‘𝐾)𝑧))
22 breq2 5152 . . . . . . 7 (𝑥 = (𝑈𝑆) → (𝑦(le‘𝐾)𝑥𝑦(le‘𝐾)(𝑈𝑆)))
2322ralbidv 3176 . . . . . 6 (𝑥 = (𝑈𝑆) → (∀𝑦𝑇 𝑦(le‘𝐾)𝑥 ↔ ∀𝑦𝑇 𝑦(le‘𝐾)(𝑈𝑆)))
24 breq1 5151 . . . . . . . 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 3176 . . . . . 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 48753 . . 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 18471 . 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 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963   class class class wbr 5148  dom cdm 5689  cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  lubclub 18367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-proset 18352  df-poset 18371  df-lub 18404
This theorem is referenced by:  lubprlem  48759
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