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Theorem meetfval 18281
Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18282 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 3462 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 meetfval.m . . 3 ∧ = (meetβ€˜πΎ)
3 fvex 6856 . . . . . . 7 (Baseβ€˜πΎ) ∈ V
4 moeq 3666 . . . . . . . 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 2733 . . . . . . 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 7910 . . . . . 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 18259 . . . . . . . . . . 11 Fun 𝐺
11 funbrfv2b 6901 . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2733 . . . . . . . . . . . . . 14 (leβ€˜πΎ) = (leβ€˜πΎ)
15 simpl 484 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom 𝐺) β†’ 𝐾 ∈ V)
16 simpr 486 . . . . . . . . . . . . . 14 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom 𝐺) β†’ {π‘₯, 𝑦} ∈ dom 𝐺)
1713, 14, 9, 15, 16glbelss 18261 . . . . . . . . . . . . 13 ((𝐾 ∈ V ∧ {π‘₯, 𝑦} ∈ dom 𝐺) β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
1817ex 414 . . . . . . . . . . . 12 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom 𝐺 β†’ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ)))
19 vex 3448 . . . . . . . . . . . . 13 π‘₯ ∈ V
20 vex 3448 . . . . . . . . . . . . 13 𝑦 ∈ V
2119, 20prss 4781 . . . . . . . . . . . 12 ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ↔ {π‘₯, 𝑦} βŠ† (Baseβ€˜πΎ))
2218, 21syl6ibr 252 . . . . . . . . . . 11 (𝐾 ∈ V β†’ ({π‘₯, 𝑦} ∈ dom 𝐺 β†’ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))))
23 eqcom 2740 . . . . . . . . . . . 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 611 . . . . . . . . . 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 1931 . . . . . . . 8 (𝐾 ∈ V β†’ βˆ€π‘§({π‘₯, 𝑦}𝐺𝑧 β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ)) ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))))
2827alrimiv 1931 . . . . . . 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 1931 . . . . . 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 7426 . . . . . 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 5282 . . . 4 (𝐾 ∈ V β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧} ∈ V)
33 fveq2 6843 . . . . . . . 8 (𝑝 = 𝐾 β†’ (glbβ€˜π‘) = (glbβ€˜πΎ))
3433, 9eqtr4di 2791 . . . . . . 7 (𝑝 = 𝐾 β†’ (glbβ€˜π‘) = 𝐺)
3534breqd 5117 . . . . . 6 (𝑝 = 𝐾 β†’ ({π‘₯, 𝑦} (glbβ€˜π‘)𝑧 ↔ {π‘₯, 𝑦}𝐺𝑧))
3635oprabbidv 7424 . . . . 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 18243 . . . . 5 meet = (𝑝 ∈ V ↦ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (glbβ€˜π‘)𝑧})
3836, 37fvmptg 6947 . . . 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 686 . . 3 (𝐾 ∈ V β†’ (meetβ€˜πΎ) = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
402, 39eqtrid 2785 . 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 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆƒ*wmo 2533  Vcvv 3444   βŠ† wss 3911  {cpr 4589   class class class wbr 5106  dom cdm 5634  Fun wfun 6491  β€˜cfv 6497  {coprab 7359  Basecbs 17088  lecple 17145  glbcglb 18204  meetcmee 18206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-oprab 7362  df-glb 18241  df-meet 18243
This theorem is referenced by:  meetfval2  18282  meet0  18300  odujoin  18302  odumeet  18304
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