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Theorem meetfval2 17705
 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 17704 . 2 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
41glbfun 17682 . . . . 5 Fun 𝐺
5 funbrfv2b 6716 . . . . 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 2765 . . . . 5 ((𝐺‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝐺‘{𝑥, 𝑦}))
87anbi2i 625 . . . 4 (({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧) ↔ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})))
96, 8bitri 278 . . 3 ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})))
109oprabbii 7221 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
113, 10eqtrdi 2809 1 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cpr 4527   class class class wbr 5036  dom cdm 5528  Fun wfun 6334  ‘cfv 6340  {coprab 7157  glbcglb 17632  meetcmee 17634 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-oprab 7160  df-glb 17664  df-meet 17666 This theorem is referenced by:  meetdm  17706  meetval  17708
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