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Theorem fnwe2lem3 38407
Description: Lemma for fnwe2 38408. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
fnwe2.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
fnwe2.f (𝜑 → (𝐹𝐴):𝐴𝐵)
fnwe2.r (𝜑𝑅 We 𝐵)
fnwe2lem3.a (𝜑𝑎𝐴)
fnwe2lem3.b (𝜑𝑏𝐴)
Assertion
Ref Expression
fnwe2lem3 (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
Distinct variable groups:   𝑦,𝑈,𝑧,𝑎,𝑏   𝑥,𝑆,𝑦,𝑎,𝑏   𝑥,𝑅,𝑦,𝑎,𝑏   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧,𝑎,𝑏   𝑥,𝐹,𝑦,𝑧,𝑎,𝑏   𝑇,𝑎,𝑏   𝐵,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐵(𝑥,𝑦,𝑧)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2lem3
StepHypRef Expression
1 orc 894 . . . . 5 ((𝐹𝑎)𝑅(𝐹𝑏) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
21adantl 474 . . . 4 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
3 fnwe2.su . . . . 5 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
4 fnwe2.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
53, 4fnwe2val 38404 . . . 4 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
62, 5sylibr 226 . . 3 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → 𝑎𝑇𝑏)
763mix1d 1436 . 2 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
8 simplr 786 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝐹𝑎) = (𝐹𝑏))
9 simpr 478 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)
108, 9jca 508 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
1110olcd 901 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
1211, 5sylibr 226 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎𝑇𝑏)
13123mix1d 1436 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
14 3mix2 1431 . . . 4 (𝑎 = 𝑏 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
1514adantl 474 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
16 simplr 786 . . . . . . . 8 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑎) = (𝐹𝑏))
1716eqcomd 2805 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑏) = (𝐹𝑎))
18 csbeq1 3731 . . . . . . . . . 10 ((𝐹𝑎) = (𝐹𝑏) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
1918adantl 474 . . . . . . . . 9 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
2019breqd 4854 . . . . . . . 8 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑏(𝐹𝑎) / 𝑧𝑆𝑎𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2120biimpa 469 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)
2217, 21jca 508 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2322olcd 901 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
243, 4fnwe2val 38404 . . . . 5 (𝑏𝑇𝑎 ↔ ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
2523, 24sylibr 226 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏𝑇𝑎)
26253mix3d 1438 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
27 fnwe2lem3.a . . . . . . 7 (𝜑𝑎𝐴)
28 fnwe2.s . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
293, 4, 28fnwe2lem1 38405 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3027, 29mpdan 679 . . . . . 6 (𝜑(𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
31 weso 5303 . . . . . 6 ((𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} → (𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3230, 31syl 17 . . . . 5 (𝜑(𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3332adantr 473 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3427adantr 473 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐴)
35 eqidd 2800 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑎))
36 fveqeq2 6420 . . . . . 6 (𝑦 = 𝑎 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑎) = (𝐹𝑎)))
3736elrab 3556 . . . . 5 (𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ (𝑎𝐴 ∧ (𝐹𝑎) = (𝐹𝑎)))
3834, 35, 37sylanbrc 579 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
39 fnwe2lem3.b . . . . . 6 (𝜑𝑏𝐴)
4039adantr 473 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐴)
41 simpr 478 . . . . . 6 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑏))
4241eqcomd 2805 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑏) = (𝐹𝑎))
43 fveqeq2 6420 . . . . . 6 (𝑦 = 𝑏 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑏) = (𝐹𝑎)))
4443elrab 3556 . . . . 5 (𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ (𝑏𝐴 ∧ (𝐹𝑏) = (𝐹𝑎)))
4540, 42, 44sylanbrc 579 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
46 solin 5256 . . . 4 (((𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ∧ (𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ∧ 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
4733, 38, 45, 46syl12anc 866 . . 3 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
4813, 15, 26, 47mpjao3dan 1557 . 2 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
49 orc 894 . . . . 5 ((𝐹𝑏)𝑅(𝐹𝑎) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
5049adantl 474 . . . 4 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
5150, 24sylibr 226 . . 3 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → 𝑏𝑇𝑎)
52513mix3d 1438 . 2 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
53 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
54 weso 5303 . . . 4 (𝑅 We 𝐵𝑅 Or 𝐵)
5553, 54syl 17 . . 3 (𝜑𝑅 Or 𝐵)
56 fvres 6430 . . . . 5 (𝑎𝐴 → ((𝐹𝐴)‘𝑎) = (𝐹𝑎))
5727, 56syl 17 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) = (𝐹𝑎))
58 fnwe2.f . . . . 5 (𝜑 → (𝐹𝐴):𝐴𝐵)
5958, 27ffvelrnd 6586 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) ∈ 𝐵)
6057, 59eqeltrrd 2879 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝐵)
61 fvres 6430 . . . . 5 (𝑏𝐴 → ((𝐹𝐴)‘𝑏) = (𝐹𝑏))
6239, 61syl 17 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) = (𝐹𝑏))
6358, 39ffvelrnd 6586 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) ∈ 𝐵)
6462, 63eqeltrrd 2879 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝐵)
65 solin 5256 . . 3 ((𝑅 Or 𝐵 ∧ ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵)) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
6655, 60, 64, 65syl12anc 866 . 2 (𝜑 → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
677, 48, 52, 66mpjao3dan 1557 1 (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874  w3o 1107   = wceq 1653  wcel 2157  {crab 3093  csb 3728   class class class wbr 4843  {copab 4905   Or wor 5232   We wwe 5270  cres 5314  wf 6097  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109
This theorem is referenced by:  fnwe2  38408
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