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Theorem eclclwwlkn0 24957
Description: An equivalence class according to  .~. (Contributed by Alexander van der Vekens, 12-Apr-2018.)
Hypotheses
Ref Expression
erclwwlkn.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
erclwwlkn.r  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
Assertion
Ref Expression
eclclwwlkn0  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  |  x  .~  y } ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    x, n, t, u, N    y, n, t, u, x    n, W   
x,  .~ , y    x, W   
x, E    x, N    x, V    x, X    x, B
Allowed substitution hints:    B( y, u, t, n)    .~ ( u, t, n)    E( y, n)    N( y)    V( y)    W( y)    X( y, u, t, n)

Proof of Theorem eclclwwlkn0
StepHypRef Expression
1 elqsg 7381 . 2  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  [ x ]  .~  ) )
2 vex 3112 . . . . 5  |-  x  e. 
_V
3 dfec2 7332 . . . . 5  |-  ( x  e.  _V  ->  [ x ]  .~  =  { y  |  x  .~  y } )
42, 3mp1i 12 . . . 4  |-  ( B  e.  X  ->  [ x ]  .~  =  { y  |  x  .~  y } )
54eqeq2d 2471 . . 3  |-  ( B  e.  X  ->  ( B  =  [ x ]  .~  <->  B  =  {
y  |  x  .~  y } ) )
65rexbidv 2968 . 2  |-  ( B  e.  X  ->  ( E. x  e.  W  B  =  [ x ]  .~  <->  E. x  e.  W  B  =  { y  |  x  .~  y } ) )
71, 6bitrd 253 1  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  |  x  .~  y } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   _Vcvv 3109   class class class wbr 4456   {copab 4514   ` cfv 5594  (class class class)co 6296   [cec 7327   /.cqs 7328   0cc0 9509   ...cfz 11697   cyclShift ccsh 12770   ClWWalksN cclwwlkn 24875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-ec 7331  df-qs 7335
This theorem is referenced by:  eclclwwlkn1  24958
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