ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eloprabi Unicode version

Theorem eloprabi 5764
Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabi.1  1st `  1st `
eloprabi.2  2nd `  1st `
eloprabi.3  2nd `
Assertion
Ref Expression
eloprabi  { <. <. ,  >. ,  >.  |  }
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)

Proof of Theorem eloprabi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . . . 6  <. <. ,  >. ,  >.  <. <. , 
>. ,  >.
21anbi1d 438 . . . . 5  <. <. ,  >. ,  >.  <. <. ,  >. ,  >.
323exbidv 1746 . . . 4  <. <. ,  >. ,  >.  <. <. ,  >. ,  >.
4 df-oprab 5459 . . . 4  { <. <. ,  >. ,  >.  |  }  {  |  <. <. ,  >. ,  >.  }
53, 4elab2g 2683 . . 3  { <. <. ,  >. ,  >.  |  }  { <. <. ,  >. ,  >.  |  }  <. <. , 
>. ,  >.
65ibi 165 . 2  { <. <. ,  >. ,  >.  |  }  <. <. ,  >. ,  >.
7 vex 2554 . . . . . . . . . . . 12 
_V
8 vex 2554 . . . . . . . . . . . 12 
_V
97, 8opex 3957 . . . . . . . . . . 11  <. ,  >.  _V
10 vex 2554 . . . . . . . . . . 11 
_V
119, 10op1std 5717 . . . . . . . . . 10  <. <. , 
>. ,  >.  1st `  <. , 
>.
1211fveq2d 5125 . . . . . . . . 9  <. <. , 
>. ,  >.  1st `  1st `  1st `  <. ,  >.
137, 8op1st 5715 . . . . . . . . 9  1st `  <. , 
>.
1412, 13syl6req 2086 . . . . . . . 8  <. <. , 
>. ,  >.  1st `  1st `
15 eloprabi.1 . . . . . . . 8  1st `  1st `
1614, 15syl 14 . . . . . . 7  <. <. , 
>. ,  >.
1711fveq2d 5125 . . . . . . . . 9  <. <. , 
>. ,  >.  2nd `  1st `  2nd `  <. ,  >.
187, 8op2nd 5716 . . . . . . . . 9  2nd `  <. , 
>.
1917, 18syl6req 2086 . . . . . . . 8  <. <. , 
>. ,  >.  2nd `  1st `
20 eloprabi.2 . . . . . . . 8  2nd `  1st `
2119, 20syl 14 . . . . . . 7  <. <. , 
>. ,  >.
229, 10op2ndd 5718 . . . . . . . . 9  <. <. , 
>. ,  >.  2nd `
2322eqcomd 2042 . . . . . . . 8  <. <. , 
>. ,  >.  2nd `
24 eloprabi.3 . . . . . . . 8  2nd `
2523, 24syl 14 . . . . . . 7  <. <. , 
>. ,  >.
2616, 21, 253bitrd 203 . . . . . 6  <. <. , 
>. ,  >.
2726biimpa 280 . . . . 5  <. <. ,  >. ,  >.
2827exlimiv 1486 . . . 4  <. <. ,  >. ,  >.
2928exlimiv 1486 . . 3  <. <. , 
>. ,  >.
3029exlimiv 1486 . 2  <. <. ,  >. ,  >.
316, 30syl 14 1  { <. <. ,  >. ,  >.  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   <.cop 3370   ` cfv 4845   {coprab 5456   1stc1st 5707   2ndc2nd 5708
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-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-13 1401  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  ax-un 4136
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-mpt 3811  df-id 4021  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-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator