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Theorem fnwe2lem3 39137
Description: Lemma for fnwe2 39138. 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 975 . . . 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 39134 . . . 4 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
51, 4sylibr 235 . . 3 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → 𝑎𝑇𝑏)
653mix1d 1329 . 2 ((𝜑 ∧ (𝐹𝑎)𝑅(𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
7 simplr 765 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝐹𝑎) = (𝐹𝑏))
8 simpr 485 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)
97, 8jca 512 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
109olcd 871 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
1110, 4sylibr 235 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → 𝑎𝑇𝑏)
12113mix1d 1329 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
13 3mix2 1324 . . . 4 (𝑎 = 𝑏 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
1413adantl 482 . . 3 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑎 = 𝑏) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
15 simplr 765 . . . . . . . 8 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑎) = (𝐹𝑏))
1615eqcomd 2801 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → (𝐹𝑏) = (𝐹𝑎))
17 csbeq1 3814 . . . . . . . . . 10 ((𝐹𝑎) = (𝐹𝑏) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
1817adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 = (𝐹𝑏) / 𝑧𝑆)
1918breqd 4973 . . . . . . . 8 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑏(𝐹𝑎) / 𝑧𝑆𝑎𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2019biimpa 477 . . . . . . 7 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)
2116, 20jca 512 . . . . . 6 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎))
2221olcd 871 . . . . 5 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
232, 3fnwe2val 39134 . . . . 5 (𝑏𝑇𝑎 ↔ ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
2422, 23sylibr 235 . . . 4 (((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) ∧ 𝑏(𝐹𝑎) / 𝑧𝑆𝑎) → 𝑏𝑇𝑎)
25243mix3d 1331 . . 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 39135 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
2926, 28mpdan 683 . . . . . 6 (𝜑(𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
30 weso 5434 . . . . . 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 481 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) / 𝑧𝑆 Or {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
33 fveqeq2 6547 . . . . 5 (𝑦 = 𝑎 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑎) = (𝐹𝑎)))
3426adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎𝐴)
35 eqidd 2796 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑎))
3633, 34, 35elrabd 3620 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑎 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
37 fveqeq2 6547 . . . . 5 (𝑦 = 𝑏 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑏) = (𝐹𝑎)))
38 fnwe2lem3.b . . . . . 6 (𝜑𝑏𝐴)
3938adantr 481 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏𝐴)
40 simpr 485 . . . . . 6 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑎) = (𝐹𝑏))
4140eqcomd 2801 . . . . 5 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝐹𝑏) = (𝐹𝑎))
4237, 39, 41elrabd 3620 . . . 4 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → 𝑏 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
43 solin 5386 . . . 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 833 . . 3 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎(𝐹𝑎) / 𝑧𝑆𝑏𝑎 = 𝑏𝑏(𝐹𝑎) / 𝑧𝑆𝑎))
4512, 14, 25, 44mpjao3dan 1424 . 2 ((𝜑 ∧ (𝐹𝑎) = (𝐹𝑏)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
46 animorrl 975 . . . 4 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → ((𝐹𝑏)𝑅(𝐹𝑎) ∨ ((𝐹𝑏) = (𝐹𝑎) ∧ 𝑏(𝐹𝑏) / 𝑧𝑆𝑎)))
4746, 23sylibr 235 . . 3 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → 𝑏𝑇𝑎)
48473mix3d 1331 . 2 ((𝜑 ∧ (𝐹𝑏)𝑅(𝐹𝑎)) → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
49 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
50 weso 5434 . . . 4 (𝑅 We 𝐵𝑅 Or 𝐵)
5149, 50syl 17 . . 3 (𝜑𝑅 Or 𝐵)
5226fvresd 6558 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) = (𝐹𝑎))
53 fnwe2.f . . . . 5 (𝜑 → (𝐹𝐴):𝐴𝐵)
5453, 26ffvelrnd 6717 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑎) ∈ 𝐵)
5552, 54eqeltrrd 2884 . . 3 (𝜑 → (𝐹𝑎) ∈ 𝐵)
5638fvresd 6558 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) = (𝐹𝑏))
5753, 38ffvelrnd 6717 . . . 4 (𝜑 → ((𝐹𝐴)‘𝑏) ∈ 𝐵)
5856, 57eqeltrrd 2884 . . 3 (𝜑 → (𝐹𝑏) ∈ 𝐵)
59 solin 5386 . . 3 ((𝑅 Or 𝐵 ∧ ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵)) → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
6051, 55, 58, 59syl12anc 833 . 2 (𝜑 → ((𝐹𝑎)𝑅(𝐹𝑏) ∨ (𝐹𝑎) = (𝐹𝑏) ∨ (𝐹𝑏)𝑅(𝐹𝑎)))
616, 45, 48, 60mpjao3dan 1424 1 (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 842  w3o 1079   = wceq 1522  wcel 2081  {crab 3109  csb 3811   class class class wbr 4962  {copab 5024   Or wor 5361   We wwe 5401  cres 5445  wf 6221  cfv 6225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233
This theorem is referenced by:  fnwe2  39138
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