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Theorem eclclwwlkn0 30505
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 7152 . 2  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  [ x ]  .~  ) )
2 vex 2975 . . . . 5  |-  x  e. 
_V
3 dfec2 7104 . . . . 5  |-  ( x  e.  _V  ->  [ x ]  .~  =  { y  |  x  .~  y } )
42, 3mp1i 12 . . . 4  |-  ( B  e.  X  ->  [ x ]  .~  =  { y  |  x  .~  y } )
54eqeq2d 2454 . . 3  |-  ( B  e.  X  ->  ( B  =  [ x ]  .~  <->  B  =  {
y  |  x  .~  y } ) )
65rexbidv 2736 . 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 965    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2716   _Vcvv 2972   class class class wbr 4292   {copab 4349   ` cfv 5418  (class class class)co 6091   [cec 7099   /.cqs 7100   0cc0 9282   ...cfz 11437   cyclShift ccsh 12425   ClWWalksN cclwwlkn 30414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-cnv 4848  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-ec 7103  df-qs 7107
This theorem is referenced by:  eclclwwlkn1  30506
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