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Theorem meetdm2 48867
Description: The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.)
Hypotheses
Ref Expression
joindm2.b 𝐵 = (Base‘𝐾)
joindm2.k (𝜑𝐾𝑉)
meetdm2.g 𝐺 = (glb‘𝐾)
meetdm2.m = (meet‘𝐾)
Assertion
Ref Expression
meetdm2 (𝜑 → (dom = (𝐵 × 𝐵) ↔ ∀𝑥𝐵𝑦𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
Distinct variable groups:   𝑥, ,𝑦   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem meetdm2
StepHypRef Expression
1 joindm2.b . . . 4 𝐵 = (Base‘𝐾)
2 meetdm2.m . . . 4 = (meet‘𝐾)
3 joindm2.k . . . 4 (𝜑𝐾𝑉)
41, 2, 3meetdmss 18438 . . 3 (𝜑 → dom ⊆ (𝐵 × 𝐵))
5 eqss 3999 . . . 4 (dom = (𝐵 × 𝐵) ↔ (dom ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ⊆ dom ))
65baib 535 . . 3 (dom ⊆ (𝐵 × 𝐵) → (dom = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ))
74, 6syl 17 . 2 (𝜑 → (dom = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ))
8 relxp 5703 . . 3 Rel (𝐵 × 𝐵)
9 ssrel 5792 . . 3 (Rel (𝐵 × 𝐵) → ((𝐵 × 𝐵) ⊆ dom ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom )))
108, 9mp1i 13 . 2 (𝜑 → ((𝐵 × 𝐵) ⊆ dom ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom )))
11 opelxp 5721 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
1211a1i 11 . . . . 5 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵)))
13 meetdm2.g . . . . . 6 𝐺 = (glb‘𝐾)
14 vex 3484 . . . . . . 7 𝑥 ∈ V
1514a1i 11 . . . . . 6 (𝜑𝑥 ∈ V)
16 vex 3484 . . . . . . 7 𝑦 ∈ V
1716a1i 11 . . . . . 6 (𝜑𝑦 ∈ V)
1813, 2, 3, 15, 17meetdef 18435 . . . . 5 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ↔ {𝑥, 𝑦} ∈ dom 𝐺))
1912, 18imbi12d 344 . . . 4 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ) ↔ ((𝑥𝐵𝑦𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
20192albidv 1923 . . 3 (𝜑 → (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
21 r2al 3195 . . 3 (∀𝑥𝐵𝑦𝐵 {𝑥, 𝑦} ∈ dom 𝐺 ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))
2220, 21bitr4di 289 . 2 (𝜑 → (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ) ↔ ∀𝑥𝐵𝑦𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
237, 10, 223bitrd 305 1 (𝜑 → (dom = (𝐵 × 𝐵) ↔ ∀𝑥𝐵𝑦𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  {cpr 4628  cop 4632   × cxp 5683  dom cdm 5685  Rel wrel 5690  cfv 6561  Basecbs 17247  glbcglb 18356  meetcmee 18358
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-glb 18392  df-meet 18394
This theorem is referenced by:  meetdm3  48868  toslat  48871
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