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Theorem meetfval2 18433
Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
meetfval.u 𝐺 = (glb‘𝐾)
meetfval.m = (meet‘𝐾)
Assertion
Ref Expression
meetfval2 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
Distinct variable groups:   𝑥,𝑦,𝑧,𝐾   𝑧,𝐺
Allowed substitution hints:   𝐺(𝑥,𝑦)   (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem meetfval2
StepHypRef Expression
1 meetfval.u . . 3 𝐺 = (glb‘𝐾)
2 meetfval.m . . 3 = (meet‘𝐾)
31, 2meetfval 18432 . 2 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
41glbfun 18410 . . . . 5 Fun 𝐺
5 funbrfv2b 6966 . . . . 5 (Fun 𝐺 → ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧)))
64, 5ax-mp 5 . . . 4 ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧))
7 eqcom 2744 . . . . 5 ((𝐺‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝐺‘{𝑥, 𝑦}))
87anbi2i 623 . . . 4 (({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧) ↔ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})))
96, 8bitri 275 . . 3 ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})))
109oprabbii 7500 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
113, 10eqtrdi 2793 1 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cpr 4628   class class class wbr 5143  dom cdm 5685  Fun wfun 6555  cfv 6561  {coprab 7432  glbcglb 18356  meetcmee 18358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-oprab 7435  df-glb 18392  df-meet 18394
This theorem is referenced by:  meetdm  18434  meetval  18436
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