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Theorem meetdef 18447
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 18446 . . . 4 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
54eleq2d 2824 . . 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 4737 . . . . 5 (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
109eleq1d 2823 . . . 4 (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑦} ∈ dom 𝐺))
11 preq2 4738 . . . . 5 (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
1211eleq1d 2823 . . . 4 (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
1310, 12opelopabg 5547 . . 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 279 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  {cpr 4632  cop 4636  {copab 5209  dom cdm 5688  cfv 6562  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:  meetval  18448  meetcl  18449  meetdmss  18450  meeteu  18453  clatl  18565  meetdm2  48766
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