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Theorem meetdef 18271
Description: Two ways to say that a meet is defined. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
meetdef.u 𝐺 = (glb‘𝐾)
meetdef.m = (meet‘𝐾)
meetdef.k (𝜑𝐾𝑉)
meetdef.x (𝜑𝑋𝑊)
meetdef.y (𝜑𝑌𝑍)
Assertion
Ref Expression
meetdef (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))

Proof of Theorem meetdef
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 meetdef.k . . 3 (𝜑𝐾𝑉)
2 meetdef.u . . . . 5 𝐺 = (glb‘𝐾)
3 meetdef.m . . . . 5 = (meet‘𝐾)
42, 3meetdm 18270 . . . 4 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
54eleq2d 2823 . . 3 (𝐾𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
61, 5syl 17 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}))
7 meetdef.x . . 3 (𝜑𝑋𝑊)
8 meetdef.y . . 3 (𝜑𝑌𝑍)
9 preq1 4692 . . . . 5 (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
109eleq1d 2822 . . . 4 (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺))
11 preq2 4693 . . . . 5 (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
1211eleq1d 2822 . . . 4 (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
1310, 12opelopabg 5493 . . 3 ((𝑋𝑊𝑌𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
147, 8, 13syl2anc 584 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺} ↔ {𝑋, 𝑌} ∈ dom 𝐺))
156, 14bitrd 278 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  {cpr 4586  cop 4590  {copab 5165  dom cdm 5631  cfv 6493  glbcglb 18191  meetcmee 18193
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-oprab 7357  df-glb 18228  df-meet 18230
This theorem is referenced by:  meetval  18272  meetcl  18273  meetdmss  18274  meeteu  18277  clatl  18389  meetdm2  46935
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