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Theorem unxpdomlem2 8321
Description: Lemma for unxpdom 8323. (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 466 . 2 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ 𝑠 = 𝑡)
3 elun1 3931 . . . . . . . . . 10 (𝑧𝑎𝑧 ∈ (𝑎𝑏))
43ad2antrl 707 . . . . . . . . 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 8320 . . . . . . . . 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 4231 . . . . . . . . 9 (𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
109ad2antrl 707 . . . . . . . 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 2805 . . . . . . 7 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
12 unxpdomlem2.1 . . . . . . . . . 10 (𝜑𝑤 ∈ (𝑎𝑏))
1312adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → 𝑤 ∈ (𝑎𝑏))
145, 6unxpdomlem1 8320 . . . . . . . . 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 4234 . . . . . . . . 9 𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
1716ad2antll 708 . . . . . . . 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 2805 . . . . . . 7 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
1911, 18eqeq12d 2786 . . . . . 6 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
2019biimpa 462 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
21 vex 3354 . . . . . 6 𝑧 ∈ V
22 vex 3354 . . . . . . 7 𝑡 ∈ V
23 vex 3354 . . . . . . 7 𝑠 ∈ V
2422, 23ifex 4295 . . . . . 6 if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
2521, 24opth 5072 . . . . 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 208 . . . 4 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤))
2726simprd 483 . . 3 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑤)
28 iftrue 4231 . . . . . . 7 (𝑧 = 𝑚 → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑡)
2927eqeq1d 2773 . . . . . . 7 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑡𝑤 = 𝑡))
3028, 29syl5ib 234 . . . . . 6 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚𝑤 = 𝑡))
31 iftrue 4231 . . . . . . 7 (𝑤 = 𝑡 → if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑛)
3226simpld 482 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑧 = if(𝑤 = 𝑡, 𝑛, 𝑚))
3332eqeq1d 2773 . . . . . . 7 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑛 ↔ if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑛))
3431, 33syl5ibr 236 . . . . . 6 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑤 = 𝑡𝑧 = 𝑛))
3530, 34syld 47 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚𝑧 = 𝑛))
36 unxpdomlem2.2 . . . . . . 7 (𝜑 → ¬ 𝑚 = 𝑛)
3736ad2antrr 705 . . . . . 6 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → ¬ 𝑚 = 𝑛)
38 equequ1 2110 . . . . . . 7 (𝑧 = 𝑚 → (𝑧 = 𝑛𝑚 = 𝑛))
3938notbid 307 . . . . . 6 (𝑧 = 𝑚 → (¬ 𝑧 = 𝑛 ↔ ¬ 𝑚 = 𝑛))
4037, 39syl5ibrcom 237 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚 → ¬ 𝑧 = 𝑛))
4135, 40pm2.65d 187 . . . 4 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → ¬ 𝑧 = 𝑚)
4241iffalsed 4236 . . 3 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → if(𝑧 = 𝑚, 𝑡, 𝑠) = 𝑠)
43 iffalse 4234 . . . . 5 𝑤 = 𝑡 → if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑚)
4432eqeq1d 2773 . . . . 5 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (𝑧 = 𝑚 ↔ if(𝑤 = 𝑡, 𝑛, 𝑚) = 𝑚))
4543, 44syl5ibr 236 . . . 4 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → (¬ 𝑤 = 𝑡𝑧 = 𝑚))
4641, 45mt3d 142 . . 3 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑤 = 𝑡)
4727, 42, 463eqtr3d 2813 . 2 (((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) ∧ (𝐹𝑧) = (𝐹𝑤)) → 𝑠 = 𝑡)
482, 47mtand 816 1 ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  cun 3721  ifcif 4225  cop 4322  cmpt 4863  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  unxpdomlem3  8322
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