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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  ,  S  Or  R  Or

Proof of Theorem isosolem
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 5404 . . 3  H 
Isom  R ,  S  ,  S  Po  R  Po
2 df-3an 886 . . . . . . 7  a  b  c  a  b  c
3 isof1o 5390 . . . . . . . . . . 11  H 
Isom  R ,  S  ,  H : -1-1-onto->
4 f1of 5069 . . . . . . . . . . 11  H : -1-1-onto->  H :
-->
5 ffvelrn 5243 . . . . . . . . . . . . 13  H : -->  a  H `  a
65ex 108 . . . . . . . . . . . 12  H : -->  a  H `  a
7 ffvelrn 5243 . . . . . . . . . . . . 13  H : -->  b  H `  b
87ex 108 . . . . . . . . . . . 12  H : -->  b  H `  b
9 ffvelrn 5243 . . . . . . . . . . . . 13  H : -->  c  H `  c
109ex 108 . . . . . . . . . . . 12  H : -->  c  H `  c
116, 8, 103anim123d 1213 . . . . . . . . . . 11  H : -->  a  b  c  H `
 a  H `
 b  H `
 c
123, 4, 113syl 17 . . . . . . . . . 10  H 
Isom  R ,  S  ,  a  b  c  H `  a  H `  b  H `
 c
1312imp 115 . . . . . . . . 9  H  Isom  R ,  S  ,  a  b  c  H `  a  H `  b  H `  c
14 breq1 3758 . . . . . . . . . . 11  H `  a  S  H `
 a S
15 breq1 3758 . . . . . . . . . . . 12  H `  a  S  H `
 a S
1615orbi1d 704 . . . . . . . . . . 11  H `  a  S  S  H `  a S  S
1714, 16imbi12d 223 . . . . . . . . . 10  H `  a  S  S  S  H `
 a S  H `  a S  S
18 breq2 3759 . . . . . . . . . . 11  H `  b  H `  a S  H `
 a S H `  b
19 breq2 3759 . . . . . . . . . . . 12  H `  b  S  S H `  b
2019orbi2d 703 . . . . . . . . . . 11  H `  b  H `  a S  S  H `  a S  S H `  b
2118, 20imbi12d 223 . . . . . . . . . 10  H `  b  H `  a S  H `  a S  S  H `  a S H `  b  H `  a S  S H `  b
22 breq2 3759 . . . . . . . . . . . 12  H `  c  H `  a S  H `
 a S H `  c
23 breq1 3758 . . . . . . . . . . . 12  H `  c  S H `  b  H `  c S H `  b
2422, 23orbi12d 706 . . . . . . . . . . 11  H `  c  H `  a S  S H `  b  H `  a S H `  c  H `  c S H `  b
2524imbi2d 219 . . . . . . . . . 10  H `  c  H `  a S H `  b  H `  a S  S H `  b  H `  a S H `  b  H `  a S H `  c  H `  c S H `  b
2617, 21, 25rspc3v 2659 . . . . . . . . 9  H `  a  H `  b  H `  c  S  S  S  H `  a S H `  b  H `  a S H `  c  H `  c S H `  b
2713, 26syl 14 . . . . . . . 8  H  Isom  R ,  S  ,  a  b  c  S  S  S  H `  a S H `  b  H `  a S H `  c  H `  c S H `  b
28 isorel 5391 . . . . . . . . . 10  H  Isom  R ,  S  ,  a  b  a R b  H `  a S H `  b
29283adantr3 1064 . . . . . . . . 9  H  Isom  R ,  S  ,  a  b  c  a R b  H `  a S H `  b
30 isorel 5391 . . . . . . . . . . 11  H  Isom  R ,  S  ,  a  c  a R c  H `  a S H `  c
31303adantr2 1063 . . . . . . . . . 10  H  Isom  R ,  S  ,  a  b  c  a R c  H `  a S H `  c
32 isorel 5391 . . . . . . . . . . . 12  H  Isom  R ,  S  ,  c  b  c R b  H `  c S H `  b
3332ancom2s 500 . . . . . . . . . . 11  H  Isom  R ,  S  ,  b  c  c R b  H `  c S H `  b
34333adantr1 1062 . . . . . . . . . 10  H  Isom  R ,  S  ,  a  b  c  c R b  H `  c S H `  b
3531, 34orbi12d 706 . . . . . . . . 9  H  Isom  R ,  S  ,  a  b  c  a R c  c R b  H `
 a S H `  c  H `  c S H `  b
3629, 35imbi12d 223 . . . . . . . 8  H  Isom  R ,  S  ,  a  b  c  a R b  a R c  c R b  H `  a S H `  b  H `  a S H `  c  H `  c S H `  b
3727, 36sylibrd 158 . . . . . . 7  H  Isom  R ,  S  ,  a  b  c  S  S  S  a R b  a R c  c R b
382, 37sylan2br 272 . . . . . 6  H  Isom  R ,  S  ,  a  b  c  S  S  S  a R b  a R c  c R b
3938anassrs 380 . . . . 5  H  Isom  R ,  S  , 
a  b  c  S  S  S  a R b  a R c  c R b
4039ralrimdva 2393 . . . 4  H  Isom  R ,  S  ,  a  b  S  S  S  c  a R b  a R c  c R b
4140ralrimdvva 2398 . . 3  H 
Isom  R ,  S  ,  S  S  S  a  b  c 
a R b  a R c  c R b
421, 41anim12d 318 . 2  H 
Isom  R ,  S  ,  S  Po  S  S  S  R  Po  a  b  c  a R b  a R c  c R b
43 df-iso 4025 . 2  S  Or  S  Po  S  S  S
44 df-iso 4025 . 2  R  Or  R  Po  a  b  c  a R b  a R c  c R b
4542, 43, 443imtr4g 194 1  H 
Isom  R ,  S  ,  S  Or  R  Or
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-onto->wf1o 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-bndl 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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