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Theorem meetval 17709
 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 17706 . . . . . 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 7173 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
76adantr 484 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌))
8 simpr 488 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → {𝑋, 𝑌} ∈ dom 𝐺)
9 eqidd 2759 . . . 4 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌}))
10 meetdef.x . . . . . 6 (𝜑𝑋𝑊)
11 meetdef.y . . . . . 6 (𝜑𝑌𝑍)
12 fvexd 6678 . . . . . 6 (𝜑 → (𝐺‘{𝑋, 𝑌}) ∈ V)
13 preq12 4631 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑥, 𝑦} = {𝑋, 𝑌})
1413eleq1d 2836 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
15143adant3 1129 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → ({𝑥, 𝑦} ∈ dom 𝐺 ↔ {𝑋, 𝑌} ∈ dom 𝐺))
16 simp3 1135 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → 𝑧 = (𝐺‘{𝑋, 𝑌}))
1713fveq2d 6667 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
18173adant3 1129 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝐺‘{𝑥, 𝑦}) = (𝐺‘{𝑋, 𝑌}))
1916, 18eqeq12d 2774 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → (𝑧 = (𝐺‘{𝑥, 𝑦}) ↔ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌})))
2015, 19anbi12d 633 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌𝑧 = (𝐺‘{𝑋, 𝑌})) → (({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦})) ↔ ({𝑋, 𝑌} ∈ dom 𝐺 ∧ (𝐺‘{𝑋, 𝑌}) = (𝐺‘{𝑋, 𝑌}))))
21 moeq 3623 . . . . . . . 8 ∃*𝑧 𝑧 = (𝐺‘{𝑥, 𝑦})
2221moani 2571 . . . . . . 7 ∃*𝑧({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))
23 eqid 2758 . . . . . . 7 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}
2420, 22, 23ovigg 7296 . . . . . 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})))
2510, 11, 12, 24syl3anc 1368 . . . . 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})))
2625adantr 484 . . . 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})))
278, 9, 26mp2and 698 . . 3 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))}𝑌) = (𝐺‘{𝑋, 𝑌}))
287, 27eqtrd 2793 . 2 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))
292, 3, 1, 10, 11meetdef 17708 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))
3029notbid 321 . . . . 5 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ dom ↔ ¬ {𝑋, 𝑌} ∈ dom 𝐺))
31 df-ov 7159 . . . . . 6 (𝑋 𝑌) = ( ‘⟨𝑋, 𝑌⟩)
32 ndmfv 6693 . . . . . 6 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → ( ‘⟨𝑋, 𝑌⟩) = ∅)
3331, 32syl5eq 2805 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ dom → (𝑋 𝑌) = ∅)
3430, 33syl6bir 257 . . . 4 (𝜑 → (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝑋 𝑌) = ∅))
3534imp 410 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = ∅)
36 ndmfv 6693 . . . 4 (¬ {𝑋, 𝑌} ∈ dom 𝐺 → (𝐺‘{𝑋, 𝑌}) = ∅)
3736adantl 485 . . 3 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝐺‘{𝑋, 𝑌}) = ∅)
3835, 37eqtr4d 2796 . 2 ((𝜑 ∧ ¬ {𝑋, 𝑌} ∈ dom 𝐺) → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))
3928, 38pm2.61dan 812 1 (𝜑 → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227  {cpr 4527  ⟨cop 4531  dom cdm 5528  ‘cfv 6340  (class class class)co 7156  {coprab 7157  glbcglb 17633  meetcmee 17635 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-glb 17665  df-meet 17667 This theorem is referenced by:  meetcl  17710  meetval2  17713  meetcomALT  17721  pmapmeet  37384  diameetN  38667  dihmeetlem2N  38910  dihmeetcN  38913  dihmeet  38954
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