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Theorem meetdm 16840
 Description: Domain of meet function for a poset-type structure. (Contributed by NM, 16-Sep-2018.)
Hypotheses
Ref Expression
meetfval.u 𝐺 = (glb‘𝐾)
meetfval.m = (meet‘𝐾)
Assertion
Ref Expression
meetdm (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
Distinct variable group:   𝑥,𝑦,𝐾
Allowed substitution hints:   𝐺(𝑥,𝑦)   (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem meetdm
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 meetfval.u . . . 4 𝐺 = (glb‘𝐾)
2 meetfval.m . . . 4 = (meet‘𝐾)
31, 2meetfval2 16839 . . 3 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
43dmeqd 5248 . 2 (𝐾𝑉 → dom = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
5 dmoprab 6639 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
6 fvex 6113 . . . . . 6 (𝐺‘{𝑥, 𝑦}) ∈ V
76isseti 3182 . . . . 5 𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
8 19.42v 1905 . . . . 5 (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})))
97, 8mpbiran2 956 . . . 4 (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝐺)
109opabbii 4649 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
115, 10eqtri 2632 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
124, 11syl6eq 2660 1 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cpr 4127  {copab 4642  dom cdm 5038  ‘cfv 5804  {coprab 6550  glbcglb 16766  meetcmee 16768 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-oprab 6553  df-glb 16798  df-meet 16800 This theorem is referenced by:  meetdef  16841  meetdmss  16844
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