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Theorem meetdm2 48766
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 18450 . . 3 (𝜑 → dom ⊆ (𝐵 × 𝐵))
5 eqss 4010 . . . 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 5706 . . 3 Rel (𝐵 × 𝐵)
9 ssrel 5794 . . 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 5724 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵))
1211a1i 11 . . . . 5 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥𝐵𝑦𝐵)))
13 meetdm2.g . . . . . 6 𝐺 = (glb‘𝐾)
14 vex 3481 . . . . . . 7 𝑥 ∈ V
1514a1i 11 . . . . . 6 (𝜑𝑥 ∈ V)
16 vex 3481 . . . . . . 7 𝑦 ∈ V
1716a1i 11 . . . . . 6 (𝜑𝑦 ∈ V)
1813, 2, 3, 15, 17meetdef 18447 . . . . 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 1920 . . 3 (𝜑 → (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
21 r2al 3192 . . 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 1534   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  {cpr 4632  cop 4636   × cxp 5686  dom cdm 5688  Rel wrel 5693  cfv 6562  Basecbs 17244  glbcglb 18367  meetcmee 18369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-oprab 7434  df-glb 18404  df-meet 18406
This theorem is referenced by:  meetdm3  48767  toslat  48770
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