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Theorem meetval 16947
Description: Meet value. Since both sides evaluate to when they don't exist, for convenience we drop the {𝑋, 𝑌} ∈ dom 𝐺 requirement. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
meetdef.u 𝐺 = (glb‘𝐾)
meetdef.m = (meet‘𝐾)
meetdef.k (𝜑𝐾𝑉)
meetdef.x (𝜑𝑋𝑊)
meetdef.y (𝜑𝑌𝑍)
Assertion
Ref Expression
meetval (𝜑 → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))

Proof of Theorem meetval
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 meetdef.k . . . . . 6 (𝜑𝐾𝑉)
2 meetdef.u . . . . . . 7 𝐺 = (glb‘𝐾)
3 meetdef.m . . . . . . 7 = (meet‘𝐾)
42, 3meetfval2 16944 . . . . . 6 (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
51, 4syl 17 . . . . 5 (𝜑 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})
65oveqd 6627 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
76adantr 481 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
8 simpr 477 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → {𝑋, 𝑌} ∈ dom 𝐺)
9 eqidd 2622 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌}))
10 meetdef.x . . . . . 6 (𝜑𝑋𝑊)
11 meetdef.y . . . . . 6 (𝜑𝑌𝑍)
12 fvex 6163 . . . . . . 7 (𝐺‘{𝑋, 𝑌}) ∈ V
1312a1i 11 . . . . . 6 (𝜑 → (𝐺‘{𝑋, 𝑌}) ∈ V)
14 preq12 4245 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
1514eleq1d 2683 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
16153adant3 1079 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
17 simp3 1061 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → 𝑧 = (𝐺‘{𝑋, 𝑌}))
1814fveq2d 6157 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
19183adant3 1079 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
2017, 19eqeq12d 2636 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝑧 = (𝐺‘{𝑥, 𝑦}) ↔ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})))
2116, 20anbi12d 746 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → (({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌}))))
22 moeq 3368 . . . . . . . 8 ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
2322moani 2524 . . . . . . 7 ∃*𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))
24 eqid 2621 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
2521, 23, 24ovigg 6741 . . . . . 6 ((𝑋𝑊𝑌𝑍 ∧ (𝐺‘{𝑋, 𝑌}) ∈ V) → (({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌})))
2610, 11, 13, 25syl3anc 1323 . . . . 5 (𝜑 → (({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌})))
2726adantr 481 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌})))
288, 9, 27mp2and 714 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌}))
297, 28eqtrd 2655 . 2 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))
302, 3, 1, 10, 11meetdef 16946 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))
3130notbid 308 . . . . 5 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ↔ ¬ {𝑋, 𝑌} ∈ dom 𝐺))
32 df-ov 6613 . . . . . 6 (𝑋 𝑌) = ( ‘⟨𝑋, 𝑌⟩)
33 ndmfv 6180 . . . . . 6 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → ( ‘⟨𝑋, 𝑌⟩) = ∅)
3432, 33syl5eq 2667 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → (𝑋 𝑌) = ∅)
3531, 34syl6bir 244 . . . 4 (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝑋 𝑌) = ∅))
3635imp 445 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = ∅)
37 ndmfv 6180 . . . 4 (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝐺‘{𝑋, 𝑌}) = ∅)
3837adantl 482 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = ∅)
3936, 38eqtr4d 2658 . 2 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))
4029, 39pm2.61dan 831 1 (𝜑 → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  c0 3896  {cpr 4155  cop 4159  dom cdm 5079  cfv 5852  (class class class)co 6610  {coprab 6611  glbcglb 16871  meetcmee 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-glb 16903  df-meet 16905
This theorem is referenced by:  meetcl  16948  meetval2  16951  meetcomALT  16959  pmapmeet  34566  diameetN  35852  dihmeetlem2N  36095  dihmeetcN  36098  dihmeet  36139
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