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Theorem unxpdomlem2 8050
 Description: Lemma for unxpdom 8052. (Contributed by Mario Carneiro, 13-Jan-2013.)
Hypotheses
Ref Expression
unxpdomlem1.1 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
unxpdomlem1.2 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
unxpdomlem2.1 (𝜑𝑤 ∈ (𝑎𝑏))
unxpdomlem2.2 (𝜑 → ¬ 𝑚 = 𝑛)
unxpdomlem2.3 (𝜑 → ¬ 𝑠 = 𝑡)
Assertion
Ref Expression
unxpdomlem2 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))
Distinct variable groups:   𝑤,𝐹,𝑧   𝑎,𝑏,𝑚,𝑛,𝑠,𝑡,𝑤,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐹(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐺(𝑥,𝑧,𝑤,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)

Proof of Theorem unxpdomlem2
StepHypRef Expression
1 unxpdomlem2.3 . . 3 (𝜑 → ¬ 𝑠 = 𝑡)
21adantr 480 . 2 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ 𝑠 = 𝑡)
3 elun1 3742 . . . . . . . . . 10 (𝑧𝑎𝑧 ∈ (𝑎𝑏))
43ad2antrl 760 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → 𝑧 ∈ (𝑎𝑏))
5 unxpdomlem1.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
6 unxpdomlem1.2 . . . . . . . . . 10 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
75, 6unxpdomlem1 8049 . . . . . . . . 9 (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
84, 7syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
9 iftrue 4042 . . . . . . . . 9 (𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
109ad2antrl 760 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
118, 10eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
12 unxpdomlem2.1 . . . . . . . . . 10 (𝜑𝑤 ∈ (𝑎𝑏))
1312adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → 𝑤 ∈ (𝑎𝑏))
145, 6unxpdomlem1 8049 . . . . . . . . 9 (𝑤 ∈ (𝑎𝑏) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
1513, 14syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
16 iffalse 4045 . . . . . . . . 9 𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
1716ad2antll 761 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
1815, 17eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
1911, 18eqeq12d 2625 . . . . . 6 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
2019biimpa 500 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
21 vex 3176 . . . . . 6 𝑧 ∈ V
22 vex 3176 . . . . . . 7 𝑡 ∈ V
23 vex 3176 . . . . . . 7 𝑠 ∈ V
2422, 23ifex 4106 . . . . . 6 if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
2521, 24opth 4871 . . . . 5 (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ ↔ (𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤))
2620, 25sylib 207 . . . 4 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤))
2726simprd 478 . . 3 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤)
28 iftrue 4042 . . . . . . 7 (𝑧 = 𝑚 → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑡)
2927eqeq1d 2612 . . . . . . 7 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑡𝑤 = 𝑡))
3028, 29syl5ib 233 . . . . . 6 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚𝑤 = 𝑡))
31 iftrue 4042 . . . . . . 7 (𝑤 = 𝑡 → if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑛)
3226simpld 474 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚))
3332eqeq1d 2612 . . . . . . 7 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑛 ↔ if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑛))
3431, 33syl5ibr 235 . . . . . 6 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑤 = 𝑡𝑧 = 𝑛))
3530, 34syld 46 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚𝑧 = 𝑛))
36 unxpdomlem2.2 . . . . . . 7 (𝜑 → ¬ 𝑚 = 𝑛)
3736ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → ¬ 𝑚 = 𝑛)
38 equequ1 1939 . . . . . . 7 (𝑧 = 𝑚 → (𝑧 = 𝑛𝑚 = 𝑛))
3938notbid 307 . . . . . 6 (𝑧 = 𝑚 → (¬ 𝑧 = 𝑛 ↔ ¬ 𝑚 = 𝑛))
4037, 39syl5ibrcom 236 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚 → ¬ 𝑧 = 𝑛))
4135, 40pm2.65d 186 . . . 4 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → ¬ 𝑧 = 𝑚)
4241iffalsed 4047 . . 3 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑠)
43 iffalse 4045 . . . . 5 𝑤 = 𝑡 → if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑚)
4432eqeq1d 2612 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚 ↔ if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑚))
4543, 44syl5ibr 235 . . . 4 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (¬ 𝑤 = 𝑡𝑧 = 𝑚))
4641, 45mt3d 139 . . 3 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑤 = 𝑡)
4727, 42, 463eqtr3d 2652 . 2 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑠 = 𝑡)
482, 47mtand 689 1 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  ifcif 4036  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  unxpdomlem3  8051
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