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Theorem eclclwwlkn1 24959
Description: An equivalence class according to  .~. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by Alexander van der Vekens, 15-Jun-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
eclclwwlkn1  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } ) )
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, W    x,  .~ , y    x, W    x, E    x, N    x, V    x, X    x, B, y   
y, W    y, X
Allowed substitution hints:    B( u, t, n)    .~ ( u, t, n)    E( y, n)    N( y)    V( y)    X( u, t, n)

Proof of Theorem eclclwwlkn1
StepHypRef Expression
1 erclwwlkn.w . . 3  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
2 erclwwlkn.r . . 3  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
31, 2eclclwwlkn0 24958 . 2  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  |  x  .~  y } ) )
41, 2erclwwlknsym 24953 . . . . . . . 8  |-  ( x  .~  y  ->  y  .~  x )
51, 2erclwwlknsym 24953 . . . . . . . 8  |-  ( y  .~  x  ->  x  .~  y )
64, 5impbii 188 . . . . . . 7  |-  ( x  .~  y  <->  y  .~  x )
76a1i 11 . . . . . 6  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( x  .~  y  <->  y  .~  x ) )
87abbidv 2593 . . . . 5  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  x  .~  y }  =  { y  |  y  .~  x } )
9 vex 3112 . . . . . . . 8  |-  y  e. 
_V
10 vex 3112 . . . . . . . 8  |-  x  e. 
_V
111, 2erclwwlkneq 24950 . . . . . . . 8  |-  ( ( y  e.  _V  /\  x  e.  _V )  ->  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
129, 10, 11mp2an 672 . . . . . . 7  |-  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) ) )
1312a1i 11 . . . . . 6  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
1413abbidv 2593 . . . . 5  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  y  .~  x }  =  { y  |  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) } )
15 3anan12 986 . . . . . . . 8  |-  ( ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) )  <->  ( x  e.  W  /\  (
y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
16 ibar 504 . . . . . . . . . 10  |-  ( x  e.  W  ->  (
( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) )  <->  ( x  e.  W  /\  (
y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) ) )
1716bicomd 201 . . . . . . . . 9  |-  ( x  e.  W  ->  (
( x  e.  W  /\  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) ) )  <-> 
( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) ) ) )
1817adantl 466 . . . . . . . 8  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( ( x  e.  W  /\  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) )  <->  ( y  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) ) ) )
1915, 18syl5bb 257 . . . . . . 7  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) )  <->  ( y  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) ) ) )
2019abbidv 2593 . . . . . 6  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) }  =  { y  |  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) } )
21 df-rab 2816 . . . . . 6  |-  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) }  =  {
y  |  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) }
2220, 21syl6eqr 2516 . . . . 5  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) }  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
238, 14, 223eqtrd 2502 . . . 4  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  x  .~  y }  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
2423eqeq2d 2471 . . 3  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( B  =  {
y  |  x  .~  y }  <->  B  =  {
y  e.  W  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) } ) )
2524rexbidva 2965 . 2  |-  ( B  e.  X  ->  ( E. x  e.  W  B  =  { y  |  x  .~  y } 
<->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) } ) )
263, 25bitrd 253 1  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   {crab 2811   _Vcvv 3109   class class class wbr 4456   {copab 4514   ` cfv 5594  (class class class)co 6296   /.cqs 7328   0cc0 9509   ...cfz 11697   cyclShift ccsh 12771   ClWWalksN cclwwlkn 24876
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-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-hash 12409  df-word 12546  df-concat 12548  df-substr 12550  df-csh 12772  df-clwwlk 24878  df-clwwlkn 24879
This theorem is referenced by:  eleclclwwlkn  24960  hashecclwwlkn1  24961  usghashecclwwlk  24962
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