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Theorem meetfval2 18445
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 18444 . 2 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
41glbfun 18422 . . . . 5 Fun 𝐺
5 funbrfv2b 6965 . . . . 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 2741 . . . . 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 7499 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
113, 10eqtrdi 2790 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 1536  wcel 2105  {cpr 4632   class class class wbr 5147  dom cdm 5688  Fun wfun 6556  cfv 6562  {coprab 7431  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:  meetdm  18446  meetval  18448
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