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Theorem meetfval 18388
Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18389 first to reduce net proof size (existence part)?
Hypotheses
Ref Expression
meetfval.u 𝐺 = (glbβ€˜πΎ)
meetfval.m ∧ = (meetβ€˜πΎ)
Assertion
Ref Expression
meetfval (𝐾 ∈ 𝑉 β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐾   𝑧,𝐺
Allowed substitution hints:   𝐺(π‘₯,𝑦)   ∧ (π‘₯,𝑦,𝑧)   𝑉(π‘₯,𝑦,𝑧)

Proof of Theorem meetfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 meetfval.m . . 3 ∧ = (meetβ€˜πΎ)
3 fvex 6915 . . . . . . 7 (Baseβ€˜πΎ) ∈ V
4 moeq 3704 . . . . . . . 8 βˆƒ*𝑧 𝑧 = (πΊβ€˜{π‘₯, 𝑦})
54a1i 11 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) β†’ βˆƒ*𝑧 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))
6 eqid 2728 . . . . . . 7 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}
73, 3, 5, 6oprabex 7988 . . . . . 6 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))} ∈ V
87a1i 11 . . . . 5 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))} ∈ V)
9 meetfval.u . . . . . . . . . . . 12 𝐺 = (glbβ€˜πΎ)
109glbfun 18366 . . . . . . . . . . 11 Fun 𝐺
11 funbrfv2b 6961 . . . . . . . . . . 11 (Fun 𝐺 β†’ ({π‘₯, 𝑦}𝐺𝑧 ↔ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ (πΊβ€˜{π‘₯, 𝑦}) = 𝑧)))
1210, 11ax-mp 5 . . . . . . . . . 10 ({π‘₯, 𝑦}𝐺𝑧 ↔ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ (πΊβ€˜{π‘₯, 𝑦}) = 𝑧))
13 eqid 2728 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2728 . . . . . . . . . . . . . 14 (leβ€˜πΎ) = (leβ€˜πΎ)
15 simpl 481 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom 𝐺) β†’ 𝐾 ∈ V)
16 simpr 483 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom 𝐺) β†’ {π‘₯, 𝑦} ∈ dom 𝐺)
1713, 14, 9, 15, 16glbelss 18368 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom 𝐺) β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
1817ex 411 . . . . . . . . . . . 12 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom 𝐺 β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ)))
19 vex 3477 . . . . . . . . . . . . 13 π‘₯ ∈ V
20 vex 3477 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4828 . . . . . . . . . . . 12 ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ↔ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
2218, 21imbitrrdi 251 . . . . . . . . . . 11 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom 𝐺 β†’ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))))
23 eqcom 2735 . . . . . . . . . . . 12 ((πΊβ€˜{π‘₯, 𝑦}) = 𝑧 ↔ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))
2423biimpi 215 . . . . . . . . . . 11 ((πΊβ€˜{π‘₯, 𝑦}) = 𝑧 β†’ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))
2522, 24anim12d1 608 . . . . . . . . . 10 (𝐾 ∈ V β†’ (({π‘₯, 𝑦} ∈ dom 𝐺 ∧ (πΊβ€˜{π‘₯, 𝑦}) = 𝑧) β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))))
2612, 25biimtrid 241 . . . . . . . . 9 (𝐾 ∈ V β†’ ({π‘₯, 𝑦}𝐺𝑧 β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))))
2726alrimiv 1922 . . . . . . . 8 (𝐾 ∈ V β†’ βˆ€π‘§({π‘₯, 𝑦}𝐺𝑧 β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))))
2827alrimiv 1922 . . . . . . 7 (𝐾 ∈ V β†’ βˆ€π‘¦βˆ€π‘§({π‘₯, 𝑦}𝐺𝑧 β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))))
2928alrimiv 1922 . . . . . 6 (𝐾 ∈ V β†’ βˆ€π‘₯βˆ€π‘¦βˆ€π‘§({π‘₯, 𝑦}𝐺𝑧 β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))))
30 ssoprab2 7495 . . . . . 6 (βˆ€π‘₯βˆ€π‘¦βˆ€π‘§({π‘₯, 𝑦}𝐺𝑧 β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))) β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧} βŠ† {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))})
3129, 30syl 17 . . . . 5 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧} βŠ† {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))})
328, 31ssexd 5328 . . . 4 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧} ∈ V)
33 fveq2 6902 . . . . . . . 8 (𝑝 = 𝐾 β†’ (glbβ€˜π‘) = (glbβ€˜πΎ))
3433, 9eqtr4di 2786 . . . . . . 7 (𝑝 = 𝐾 β†’ (glbβ€˜π‘) = 𝐺)
3534breqd 5163 . . . . . 6 (𝑝 = 𝐾 β†’ ({π‘₯, 𝑦} (glbβ€˜π‘)𝑧 ↔ {π‘₯, 𝑦}𝐺𝑧))
3635oprabbidv 7493 . . . . 5 (𝑝 = 𝐾 β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (glbβ€˜π‘)𝑧} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
37 df-meet 18350 . . . . 5 meet = (𝑝 ∈ V ↦ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (glbβ€˜π‘)𝑧})
3836, 37fvmptg 7008 . . . 4 ((𝐾 ∈ V ∧ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧} ∈ V) β†’ (meetβ€˜πΎ) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
3932, 38mpdan 685 . . 3 (𝐾 ∈ V β†’ (meetβ€˜πΎ) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
402, 39eqtrid 2780 . 2 (𝐾 ∈ V β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
411, 40syl 17 1 (𝐾 ∈ 𝑉 β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆƒ*wmo 2527  Vcvv 3473   βŠ† wss 3949  {cpr 4634   class class class wbr 5152  dom cdm 5682  Fun wfun 6547  β€˜cfv 6553  {coprab 7427  Basecbs 17189  lecple 17249  glbcglb 18311  meetcmee 18313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-oprab 7430  df-glb 18348  df-meet 18350
This theorem is referenced by:  meetfval2  18389  meet0  18407  odujoin  18409  odumeet  18411
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