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Theorem meetval 18150
Description: Meet value. Since both sides evaluate to βˆ… when they don't exist, for convenience we drop the {𝑋, π‘Œ} ∈ dom 𝐺 requirement. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
meetdef.u 𝐺 = (glbβ€˜πΎ)
meetdef.m ∧ = (meetβ€˜πΎ)
meetdef.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
meetdef.x (πœ‘ β†’ 𝑋 ∈ π‘Š)
meetdef.y (πœ‘ β†’ π‘Œ ∈ 𝑍)
Assertion
Ref Expression
meetval (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (πΊβ€˜{𝑋, π‘Œ}))

Proof of Theorem meetval
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 meetdef.k . . . . . 6 (πœ‘ β†’ 𝐾 ∈ 𝑉)
2 meetdef.u . . . . . . 7 𝐺 = (glbβ€˜πΎ)
3 meetdef.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
42, 3meetfval2 18147 . . . . . 6 (𝐾 ∈ 𝑉 β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))})
51, 4syl 17 . . . . 5 (πœ‘ β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))})
65oveqd 7320 . . . 4 (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}π‘Œ))
76adantr 482 . . 3 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (𝑋 ∧ π‘Œ) = (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}π‘Œ))
8 simpr 486 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ {𝑋, π‘Œ} ∈ dom 𝐺)
9 eqidd 2737 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (πΊβ€˜{𝑋, π‘Œ}) = (πΊβ€˜{𝑋, π‘Œ}))
10 meetdef.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ π‘Š)
11 meetdef.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑍)
12 fvexd 6815 . . . . . 6 (πœ‘ β†’ (πΊβ€˜{𝑋, π‘Œ}) ∈ V)
13 preq12 4675 . . . . . . . . . 10 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ {π‘₯, 𝑦} = {𝑋, π‘Œ})
1413eleq1d 2821 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ({π‘₯, 𝑦} ∈ dom 𝐺 ↔ {𝑋, π‘Œ} ∈ dom 𝐺))
15143adant3 1132 . . . . . . . 8 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (πΊβ€˜{𝑋, π‘Œ})) β†’ ({π‘₯, 𝑦} ∈ dom 𝐺 ↔ {𝑋, π‘Œ} ∈ dom 𝐺))
16 simp3 1138 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (πΊβ€˜{𝑋, π‘Œ})) β†’ 𝑧 = (πΊβ€˜{𝑋, π‘Œ}))
1713fveq2d 6804 . . . . . . . . . 10 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (πΊβ€˜{π‘₯, 𝑦}) = (πΊβ€˜{𝑋, π‘Œ}))
18173adant3 1132 . . . . . . . . 9 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (πΊβ€˜{𝑋, π‘Œ})) β†’ (πΊβ€˜{π‘₯, 𝑦}) = (πΊβ€˜{𝑋, π‘Œ}))
1916, 18eqeq12d 2752 . . . . . . . 8 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (πΊβ€˜{𝑋, π‘Œ})) β†’ (𝑧 = (πΊβ€˜{π‘₯, 𝑦}) ↔ (πΊβ€˜{𝑋, π‘Œ}) = (πΊβ€˜{𝑋, π‘Œ})))
2015, 19anbi12d 632 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ ∧ 𝑧 = (πΊβ€˜{𝑋, π‘Œ})) β†’ (({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦})) ↔ ({𝑋, π‘Œ} ∈ dom 𝐺 ∧ (πΊβ€˜{𝑋, π‘Œ}) = (πΊβ€˜{𝑋, π‘Œ}))))
21 moeq 3647 . . . . . . . 8 βˆƒ*𝑧 𝑧 = (πΊβ€˜{π‘₯, 𝑦})
2221moani 2551 . . . . . . 7 βˆƒ*𝑧({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))
23 eqid 2736 . . . . . . 7 {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}
2420, 22, 23ovigg 7446 . . . . . 6 ((𝑋 ∈ π‘Š ∧ π‘Œ ∈ 𝑍 ∧ (πΊβ€˜{𝑋, π‘Œ}) ∈ V) β†’ (({𝑋, π‘Œ} ∈ dom 𝐺 ∧ (πΊβ€˜{𝑋, π‘Œ}) = (πΊβ€˜{𝑋, π‘Œ})) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}π‘Œ) = (πΊβ€˜{𝑋, π‘Œ})))
2510, 11, 12, 24syl3anc 1371 . . . . 5 (πœ‘ β†’ (({𝑋, π‘Œ} ∈ dom 𝐺 ∧ (πΊβ€˜{𝑋, π‘Œ}) = (πΊβ€˜{𝑋, π‘Œ})) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}π‘Œ) = (πΊβ€˜{𝑋, π‘Œ})))
2625adantr 482 . . . 4 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (({𝑋, π‘Œ} ∈ dom 𝐺 ∧ (πΊβ€˜{𝑋, π‘Œ}) = (πΊβ€˜{𝑋, π‘Œ})) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}π‘Œ) = (πΊβ€˜{𝑋, π‘Œ})))
278, 9, 26mp2and 697 . . 3 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (𝑋{⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))}π‘Œ) = (πΊβ€˜{𝑋, π‘Œ}))
287, 27eqtrd 2776 . 2 ((πœ‘ ∧ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (𝑋 ∧ π‘Œ) = (πΊβ€˜{𝑋, π‘Œ}))
292, 3, 1, 10, 11meetdef 18149 . . . . . 6 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ ↔ {𝑋, π‘Œ} ∈ dom 𝐺))
3029notbid 319 . . . . 5 (πœ‘ β†’ (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ ↔ Β¬ {𝑋, π‘Œ} ∈ dom 𝐺))
31 df-ov 7306 . . . . . 6 (𝑋 ∧ π‘Œ) = ( ∧ β€˜βŸ¨π‘‹, π‘ŒβŸ©)
32 ndmfv 6832 . . . . . 6 (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ β†’ ( ∧ β€˜βŸ¨π‘‹, π‘ŒβŸ©) = βˆ…)
3331, 32eqtrid 2788 . . . . 5 (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ β†’ (𝑋 ∧ π‘Œ) = βˆ…)
3430, 33syl6bir 255 . . . 4 (πœ‘ β†’ (Β¬ {𝑋, π‘Œ} ∈ dom 𝐺 β†’ (𝑋 ∧ π‘Œ) = βˆ…))
3534imp 408 . . 3 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (𝑋 ∧ π‘Œ) = βˆ…)
36 ndmfv 6832 . . . 4 (Β¬ {𝑋, π‘Œ} ∈ dom 𝐺 β†’ (πΊβ€˜{𝑋, π‘Œ}) = βˆ…)
3736adantl 483 . . 3 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (πΊβ€˜{𝑋, π‘Œ}) = βˆ…)
3835, 37eqtr4d 2779 . 2 ((πœ‘ ∧ Β¬ {𝑋, π‘Œ} ∈ dom 𝐺) β†’ (𝑋 ∧ π‘Œ) = (πΊβ€˜{𝑋, π‘Œ}))
3928, 38pm2.61dan 811 1 (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (πΊβ€˜{𝑋, π‘Œ}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  Vcvv 3437  βˆ…c0 4262  {cpr 4567  βŸ¨cop 4571  dom cdm 5596  β€˜cfv 6454  (class class class)co 7303  {coprab 7304  glbcglb 18069  meetcmee 18071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-glb 18106  df-meet 18108
This theorem is referenced by:  meetcl  18151  meetval2  18154  meetcomALT  18162  pmapmeet  37826  diameetN  39109  dihmeetlem2N  39352  dihmeetcN  39355  dihmeet  39396  posmidm  46324  toplatmeet  46346
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