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Theorem fnwe2lem3 43069
Description: Lemma for fnwe2 43070. 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 animorrl 982 . . . 4 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
2 fnwe2.su . . . . 5 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
3 fnwe2.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
42, 3fnwe2val 43066 . . . 4 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
51, 4sylibr 234 . . 3 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → 𝑎𝑇𝑏)
653mix1d 1336 . 2 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
7 simplr 768 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝐹𝑎) = (𝐹𝑏))
8 simpr 484 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)
97, 8jca 511 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
109olcd 874 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
1110, 4sylibr 234 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎𝑇𝑏)
12113mix1d 1336 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
13 3mix2 1331 . . . 4 (𝑎 = 𝑏 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
1413adantl 481 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
15 simplr 768 . . . . . . . 8 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑎) = (𝐹𝑏))
1615eqcomd 2742 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑏) = (𝐹𝑎))
17 csbeq1 3901 . . . . . . . . . 10 ((𝐹𝑎) = (𝐹𝑏) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
1817adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
1918breqd 5153 . . . . . . . 8 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑏(𝐹𝑎) / 𝑧𝑆𝑎𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2019biimpa 476 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)
2116, 20jca 511 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2221olcd 874 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
232, 3fnwe2val 43066 . . . . 5 (𝑏𝑇𝑎 ↔ ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
2422, 23sylibr 234 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏𝑇𝑎)
25243mix3d 1338 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
26 fnwe2lem3.a . . . . . . 7 (𝜑𝑎𝐴)
27 fnwe2.s . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
282, 3, 27fnwe2lem1 43067 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
2926, 28mpdan 687 . . . . . 6 (𝜑(𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
30 weso 5675 . . . . . 6 ((𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} → (𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3129, 30syl 17 . . . . 5 (𝜑(𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
3231adantr 480 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
33 fveqeq2 6914 . . . . 5 (𝑦 = 𝑎 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑎) = (𝐹𝑎)))
3426adantr 480 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐴)
35 eqidd 2737 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑎))
3633, 34, 35elrabd 3693 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
37 fveqeq2 6914 . . . . 5 (𝑦 = 𝑏 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑏) = (𝐹𝑎)))
38 fnwe2lem3.b . . . . . 6 (𝜑𝑏𝐴)
3938adantr 480 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐴)
40 simpr 484 . . . . . 6 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑏))
4140eqcomd 2742 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑏) = (𝐹𝑎))
4237, 39, 41elrabd 3693 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
43 solin 5618 . . . 4 (((𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ∧ (𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ∧ 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
4432, 36, 42, 43syl12anc 836 . . 3 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
4512, 14, 25, 44mpjao3dan 1433 . 2 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
46 animorrl 982 . . . 4 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
4746, 23sylibr 234 . . 3 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → 𝑏𝑇𝑎)
48473mix3d 1338 . 2 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
49 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
50 weso 5675 . . . 4 (𝑅 We 𝐵𝑅 Or 𝐵)
5149, 50syl 17 . . 3 (𝜑𝑅 Or 𝐵)
5226fvresd 6925 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) = (𝐹𝑎))
53 fnwe2.f . . . . 5 (𝜑 → (𝐹𝐴):𝐴𝐵)
5453, 26ffvelcdmd 7104 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) ∈ 𝐵)
5552, 54eqeltrrd 2841 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝐵)
5638fvresd 6925 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) = (𝐹𝑏))
5753, 38ffvelcdmd 7104 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) ∈ 𝐵)
5856, 57eqeltrrd 2841 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝐵)
59 solin 5618 . . 3 ((𝑅 Or 𝐵 ∧ ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵)) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
6051, 55, 58, 59syl12anc 836 . 2 (𝜑 → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
616, 45, 48, 60mpjao3dan 1433 1 (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3o 1085   = wceq 1539  wcel 2107  {crab 3435  csb 3898   class class class wbr 5142  {copab 5204   Or wor 5590   We wwe 5635  cres 5686  wf 6556  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568
This theorem is referenced by:  fnwe2  43070
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