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Theorem joindm3 48866
Description: The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
Hypotheses
Ref Expression
joindm2.b 𝐵 = (Base‘𝐾)
joindm2.k (𝜑𝐾𝑉)
joindm2.u 𝑈 = (lub‘𝐾)
joindm2.j = (join‘𝐾)
joindm3.l = (le‘𝐾)
Assertion
Ref Expression
joindm3 (𝜑 → (dom = (𝐵 × 𝐵) ↔ ∀𝑥𝐵𝑦𝐵 ∃!𝑧𝐵 ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐾,𝑧   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑈(𝑥,𝑦,𝑧,𝑤)   𝐾(𝑥,𝑦)   (𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem joindm3
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 joindm2.b . . 3 𝐵 = (Base‘𝐾)
2 joindm2.k . . 3 (𝜑𝐾𝑉)
3 joindm2.u . . 3 𝑈 = (lub‘𝐾)
4 joindm2.j . . 3 = (join‘𝐾)
51, 2, 3, 4joindm2 48865 . 2 (𝜑 → (dom = (𝐵 × 𝐵) ↔ ∀𝑥𝐵𝑦𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
6 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
7 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
86, 7prssd 4822 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → {𝑥, 𝑦} ⊆ 𝐵)
9 joindm3.l . . . . . . 7 = (le‘𝐾)
10 biid 261 . . . . . . 7 ((∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤)) ↔ (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤)))
111, 9, 3, 10, 2lubeldm 18398 . . . . . 6 (𝜑 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃!𝑧𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤)))))
1211baibd 539 . . . . 5 ((𝜑 ∧ {𝑥, 𝑦} ⊆ 𝐵) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ∃!𝑧𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤))))
138, 12syldan 591 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ∃!𝑧𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤))))
142adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐾𝑉)
151, 9, 4, 14, 6, 7joinval2lem 18425 . . . . . 6 ((𝑥𝐵𝑦𝐵) → ((∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤)) ↔ ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
1615reubidv 3398 . . . . 5 ((𝑥𝐵𝑦𝐵) → (∃!𝑧𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤)) ↔ ∃!𝑧𝐵 ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
1716adantl 481 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (∃!𝑧𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑧 ∧ ∀𝑤𝐵 (∀𝑣 ∈ {𝑥, 𝑦}𝑣 𝑤𝑧 𝑤)) ↔ ∃!𝑧𝐵 ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
1813, 17bitrd 279 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ ∃!𝑧𝐵 ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
19182ralbidva 3219 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵 {𝑥, 𝑦} ∈ dom 𝑈 ↔ ∀𝑥𝐵𝑦𝐵 ∃!𝑧𝐵 ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
205, 19bitrd 279 1 (𝜑 → (dom = (𝐵 × 𝐵) ↔ ∀𝑥𝐵𝑦𝐵 ∃!𝑧𝐵 ((𝑥 𝑧𝑦 𝑧) ∧ ∀𝑤𝐵 ((𝑥 𝑤𝑦 𝑤) → 𝑧 𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  ∃!wreu 3378  wss 3951  {cpr 4628   class class class wbr 5143   × cxp 5683  dom cdm 5685  cfv 6561  Basecbs 17247  lecple 17304  lubclub 18355  joincjn 18357
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  ax-un 7755
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-oprab 7435  df-lub 18391  df-join 18393
This theorem is referenced by: (None)
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