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Theorem isosolem 5406
Description: Lemma for isoso 5407. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isosolem (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Or B𝑅 Or A))

Proof of Theorem isosolem
Dummy variables 𝑎 𝑏 𝑐 x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 5404 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Po B𝑅 Po A))
2 df-3an 886 . . . . . . 7 ((𝑎 A 𝑏 A 𝑐 A) ↔ ((𝑎 A 𝑏 A) 𝑐 A))
3 isof1o 5390 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
4 f1of 5069 . . . . . . . . . . 11 (𝐻:A1-1-ontoB𝐻:AB)
5 ffvelrn 5243 . . . . . . . . . . . . 13 ((𝐻:AB 𝑎 A) → (𝐻𝑎) B)
65ex 108 . . . . . . . . . . . 12 (𝐻:AB → (𝑎 A → (𝐻𝑎) B))
7 ffvelrn 5243 . . . . . . . . . . . . 13 ((𝐻:AB 𝑏 A) → (𝐻𝑏) B)
87ex 108 . . . . . . . . . . . 12 (𝐻:AB → (𝑏 A → (𝐻𝑏) B))
9 ffvelrn 5243 . . . . . . . . . . . . 13 ((𝐻:AB 𝑐 A) → (𝐻𝑐) B)
109ex 108 . . . . . . . . . . . 12 (𝐻:AB → (𝑐 A → (𝐻𝑐) B))
116, 8, 103anim123d 1213 . . . . . . . . . . 11 (𝐻:AB → ((𝑎 A 𝑏 A 𝑐 A) → ((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B)))
123, 4, 113syl 17 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (A, B) → ((𝑎 A 𝑏 A 𝑐 A) → ((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B)))
1312imp 115 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → ((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B))
14 breq1 3758 . . . . . . . . . . 11 (x = (𝐻𝑎) → (x𝑆y ↔ (𝐻𝑎)𝑆y))
15 breq1 3758 . . . . . . . . . . . 12 (x = (𝐻𝑎) → (x𝑆z ↔ (𝐻𝑎)𝑆z))
1615orbi1d 704 . . . . . . . . . . 11 (x = (𝐻𝑎) → ((x𝑆z z𝑆y) ↔ ((𝐻𝑎)𝑆z z𝑆y)))
1714, 16imbi12d 223 . . . . . . . . . 10 (x = (𝐻𝑎) → ((x𝑆y → (x𝑆z z𝑆y)) ↔ ((𝐻𝑎)𝑆y → ((𝐻𝑎)𝑆z z𝑆y))))
18 breq2 3759 . . . . . . . . . . 11 (y = (𝐻𝑏) → ((𝐻𝑎)𝑆y ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
19 breq2 3759 . . . . . . . . . . . 12 (y = (𝐻𝑏) → (z𝑆yz𝑆(𝐻𝑏)))
2019orbi2d 703 . . . . . . . . . . 11 (y = (𝐻𝑏) → (((𝐻𝑎)𝑆z z𝑆y) ↔ ((𝐻𝑎)𝑆z z𝑆(𝐻𝑏))))
2118, 20imbi12d 223 . . . . . . . . . 10 (y = (𝐻𝑏) → (((𝐻𝑎)𝑆y → ((𝐻𝑎)𝑆z z𝑆y)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆z z𝑆(𝐻𝑏)))))
22 breq2 3759 . . . . . . . . . . . 12 (z = (𝐻𝑐) → ((𝐻𝑎)𝑆z ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
23 breq1 3758 . . . . . . . . . . . 12 (z = (𝐻𝑐) → (z𝑆(𝐻𝑏) ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
2422, 23orbi12d 706 . . . . . . . . . . 11 (z = (𝐻𝑐) → (((𝐻𝑎)𝑆z z𝑆(𝐻𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏))))
2524imbi2d 219 . . . . . . . . . 10 (z = (𝐻𝑐) → (((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆z z𝑆(𝐻𝑏))) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
2617, 21, 25rspc3v 2659 . . . . . . . . 9 (((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
2713, 26syl 14 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
28 isorel 5391 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
29283adantr3 1064 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
30 isorel 5391 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑐 A)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
31303adantr2 1063 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
32 isorel 5391 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑐 A 𝑏 A)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3332ancom2s 500 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑏 A 𝑐 A)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
34333adantr1 1062 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3531, 34orbi12d 706 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → ((𝑎𝑅𝑐 𝑐𝑅𝑏) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏))))
3629, 35imbi12d 223 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
3727, 36sylibrd 158 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
382, 37sylan2br 272 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (A, B) ((𝑎 A 𝑏 A) 𝑐 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
3938anassrs 380 . . . . 5 (((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A)) 𝑐 A) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
4039ralrimdva 2393 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
4140ralrimdvva 2398 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
421, 41anim12d 318 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) → ((𝑆 Po B x B y B z B (x𝑆y → (x𝑆z z𝑆y))) → (𝑅 Po A 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏)))))
43 df-iso 4025 . 2 (𝑆 Or B ↔ (𝑆 Po B x B y B z B (x𝑆y → (x𝑆z z𝑆y))))
44 df-iso 4025 . 2 (𝑅 Or A ↔ (𝑅 Po A 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
4542, 43, 443imtr4g 194 1 (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Or B𝑅 Or A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  wral 2300   class class class wbr 3755   Po wpo 4022   Or wor 4023  wf 4841  1-1-ontowf1o 4844  cfv 4845   Isom wiso 4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-po 4024  df-iso 4025  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by:  isoso  5407
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