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Theorem fnwe2lem3 37102
Description: Lemma for fnwe2 37103. 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 400 . . . . 5 ((𝐹𝑎)𝑅(𝐹𝑏) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
21adantl 482 . . . 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 37099 . . . 4 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
62, 5sylibr 224 . . 3 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → 𝑎𝑇𝑏)
763mix1d 1234 . 2 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
8 simplr 791 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝐹𝑎) = (𝐹𝑏))
9 simpr 477 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)
108, 9jca 554 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
1110olcd 408 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
1211, 5sylibr 224 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎𝑇𝑏)
13123mix1d 1234 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
14 3mix2 1229 . . . 4 (𝑎 = 𝑏 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
1514adantl 482 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
16 simplr 791 . . . . . . . 8 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑎) = (𝐹𝑏))
1716eqcomd 2627 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑏) = (𝐹𝑎))
18 csbeq1 3517 . . . . . . . . . 10 ((𝐹𝑎) = (𝐹𝑏) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
1918adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
2019breqd 4624 . . . . . . . 8 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑏(𝐹𝑎) / 𝑧𝑆𝑎𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2120biimpa 501 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)
2217, 21jca 554 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2322olcd 408 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
243, 4fnwe2val 37099 . . . . 5 (𝑏𝑇𝑎 ↔ ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
2523, 24sylibr 224 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏𝑇𝑎)
26253mix3d 1236 . . 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 37100 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3027, 29mpdan 701 . . . . . 6 (𝜑(𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
31 weso 5065 . . . . . 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 481 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3427adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐴)
35 eqidd 2622 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑎))
36 fveq2 6148 . . . . . . 7 (𝑦 = 𝑎 → (𝐹𝑦) = (𝐹𝑎))
3736eqeq1d 2623 . . . . . 6 (𝑦 = 𝑎 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑎) = (𝐹𝑎)))
3837elrab 3346 . . . . 5 (𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ (𝑎𝐴 ∧ (𝐹𝑎) = (𝐹𝑎)))
3934, 35, 38sylanbrc 697 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
40 fnwe2lem3.b . . . . . 6 (𝜑𝑏𝐴)
4140adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐴)
42 simpr 477 . . . . . 6 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑏))
4342eqcomd 2627 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑏) = (𝐹𝑎))
44 fveq2 6148 . . . . . . 7 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
4544eqeq1d 2623 . . . . . 6 (𝑦 = 𝑏 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑏) = (𝐹𝑎)))
4645elrab 3346 . . . . 5 (𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ (𝑏𝐴 ∧ (𝐹𝑏) = (𝐹𝑎)))
4741, 43, 46sylanbrc 697 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
48 solin 5018 . . . 4 (((𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ∧ (𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ∧ 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
4933, 39, 47, 48syl12anc 1321 . . 3 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
5013, 15, 26, 49mpjao3dan 1392 . 2 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
51 orc 400 . . . . 5 ((𝐹𝑏)𝑅(𝐹𝑎) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
5251adantl 482 . . . 4 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
5352, 24sylibr 224 . . 3 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → 𝑏𝑇𝑎)
54533mix3d 1236 . 2 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
55 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
56 weso 5065 . . . 4 (𝑅 We 𝐵𝑅 Or 𝐵)
5755, 56syl 17 . . 3 (𝜑𝑅 Or 𝐵)
58 fvres 6164 . . . . 5 (𝑎𝐴 → ((𝐹𝐴)‘𝑎) = (𝐹𝑎))
5927, 58syl 17 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) = (𝐹𝑎))
60 fnwe2.f . . . . 5 (𝜑 → (𝐹𝐴):𝐴𝐵)
6160, 27ffvelrnd 6316 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) ∈ 𝐵)
6259, 61eqeltrrd 2699 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝐵)
63 fvres 6164 . . . . 5 (𝑏𝐴 → ((𝐹𝐴)‘𝑏) = (𝐹𝑏))
6440, 63syl 17 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) = (𝐹𝑏))
6560, 40ffvelrnd 6316 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) ∈ 𝐵)
6664, 65eqeltrrd 2699 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝐵)
67 solin 5018 . . 3 ((𝑅 Or 𝐵 ∧ ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵)) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
6857, 62, 66, 67syl12anc 1321 . 2 (𝜑 → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
697, 50, 54, 68mpjao3dan 1392 1 (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3o 1035   = wceq 1480  wcel 1987  {crab 2911  csb 3514   class class class wbr 4613  {copab 4672   Or wor 4994   We wwe 5032  cres 5076  wf 5843  cfv 5847
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855
This theorem is referenced by:  fnwe2  37103
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