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Theorem meetfval2 17331
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 17330 . 2 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
41glbfun 17308 . . . . 5 Fun 𝐺
5 funbrfv2b 6465 . . . . 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 2806 . . . . 5 ((𝐺‘{𝑥, 𝑦}) = 𝑧𝑧 = (𝐺‘{𝑥, 𝑦}))
87anbi2i 617 . . . 4 (({𝑥, 𝑦} ∈ dom 𝐺 ∧ (𝐺‘{𝑥, 𝑦}) = 𝑧) ↔ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})))
96, 8bitri 267 . . 3 ({𝑥, 𝑦}𝐺𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})))
109oprabbii 6944 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
113, 10syl6eq 2849 1 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  {cpr 4370   class class class wbr 4843  dom cdm 5312  Fun wfun 6095  cfv 6101  {coprab 6879  glbcglb 17258  meetcmee 17260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-oprab 6882  df-glb 17290  df-meet 17292
This theorem is referenced by:  meetdm  17332  meetval  17334
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