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Theorem meetdm 17623
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 17622 . . 3 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
43dmeqd 5761 . 2 (𝐾𝑉 → dom = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
5 dmoprab 7244 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
6 fvex 6671 . . . . . 6 (𝐺‘{𝑥, 𝑦}) ∈ V
76isseti 3494 . . . . 5 𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
8 19.42v 1955 . . . . 5 (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ ∃𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})))
97, 8mpbiran2 709 . . . 4 (∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ {𝑥, 𝑦} ∈ dom 𝐺)
109opabbii 5119 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
115, 10eqtri 2847 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺}
124, 11syl6eq 2875 1 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  {cpr 4551  {copab 5114  dom cdm 5542  cfv 6343  {coprab 7146  glbcglb 17549  meetcmee 17551
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
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 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-oprab 7149  df-glb 17581  df-meet 17583
This theorem is referenced by:  meetdef  17624  meetdmss  17627
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