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Theorem meetdmss 17634
Description: Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetdmss.b 𝐵 = (Base‘𝐾)
meetdmss.j = (meet‘𝐾)
meetdmss.k (𝜑𝐾𝑉)
Assertion
Ref Expression
meetdmss (𝜑 → dom ⊆ (𝐵 × 𝐵))

Proof of Theorem meetdmss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5684 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}
2 meetdmss.k . . . . 5 (𝜑𝐾𝑉)
3 eqid 2824 . . . . . 6 (glb‘𝐾) = (glb‘𝐾)
4 meetdmss.j . . . . . 6 = (meet‘𝐾)
53, 4meetdm 17630 . . . . 5 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})
62, 5syl 17 . . . 4 (𝜑 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})
76releqd 5641 . . 3 (𝜑 → (Rel dom ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}))
81, 7mpbiri 261 . 2 (𝜑 → Rel dom )
9 vex 3484 . . . . 5 𝑥 ∈ V
109a1i 11 . . . 4 (𝜑𝑥 ∈ V)
11 vex 3484 . . . . 5 𝑦 ∈ V
1211a1i 11 . . . 4 (𝜑𝑦 ∈ V)
133, 4, 2, 10, 12meetdef 17631 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
14 meetdmss.b . . . . . 6 𝐵 = (Base‘𝐾)
15 eqid 2824 . . . . . 6 (le‘𝐾) = (le‘𝐾)
162adantr 484 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → 𝐾𝑉)
17 simpr 488 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾))
1814, 15, 3, 16, 17glbelss 17608 . . . . 5 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
1918ex 416 . . . 4 (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
209, 11prss 4738 . . . . 5 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21 opelxpi 5580 . . . . 5 ((𝑥𝐵𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2220, 21sylbir 238 . . . 4 ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2319, 22syl6 35 . . 3 (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
2413, 23sylbid 243 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
258, 24relssdv 5649 1 (𝜑 → dom ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3481  wss 3920  {cpr 4553  cop 4557  {copab 5115   × cxp 5541  dom cdm 5543  Rel wrel 5548  cfv 6344  Basecbs 16486  lecple 16575  glbcglb 17556  meetcmee 17558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-oprab 7154  df-glb 17588  df-meet 17590
This theorem is referenced by:  clatl  17729
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