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Theorem clwwlkbij 30610
Description: There is a bijection between the set of closed walks of a fixed length represented by walks (as word) and the set of closed walks (as words) of a fixed length. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
Assertion
Ref Expression
clwwlkbij  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  E. f  f : { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
 N ) )
Distinct variable groups:    f, E, w    f, N, w    f, V, w
Allowed substitution hints:    X( w, f)    Y( w, f)

Proof of Theorem clwwlkbij
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvex 5810 . . . 4  |-  ( ( V WWalksN  E ) `  N
)  e.  _V
21rabex 4552 . . 3  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  e.  _V
32mptex 6058 . 2  |-  ( x  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  |->  ( x substr  <. 0 ,  N >. ) )  e.  _V
4 eqid 2454 . . 3  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  =  { w  e.  (
( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }
5 eqid 2454 . . 3  |-  ( x  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  |->  ( x substr  <. 0 ,  N >. ) )  =  ( x  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  |->  ( x substr  <. 0 ,  N >. ) )
64, 5clwwlkf1o 30609 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  ( x  e.  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( lastS  `  w
)  =  ( w `
 0 ) } 
|->  ( x substr  <. 0 ,  N >. ) ) : { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
 N ) )
7 f1oeq1 5741 . . 3  |-  ( f  =  ( x  e. 
{ w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }  |->  ( x substr  <. 0 ,  N >. ) )  -> 
( f : {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( lastS  `  w
)  =  ( w `
 0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `  N )  <->  ( x  e.  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }  |->  ( x substr  <. 0 ,  N >. ) ) : { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
 N ) ) )
87spcegv 3164 . 2  |-  ( ( x  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  |->  ( x substr  <. 0 ,  N >. ) )  e.  _V  ->  ( ( x  e. 
{ w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }  |->  ( x substr  <. 0 ,  N >. ) ) : { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
 N )  ->  E. f  f : { w  e.  (
( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
 N ) ) )
93, 6, 8mpsyl 63 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  E. f  f : { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {crab 2803   _Vcvv 3078   <.cop 3992    |-> cmpt 4459   -1-1-onto->wf1o 5526   ` cfv 5527  (class class class)co 6201   0cc0 9394   NNcn 10434   lastS clsw 12341   substr csubstr 12344   WWalksN cwwlkn 30461   ClWWalksN cclwwlkn 30563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-lsw 12349  df-concat 12350  df-s1 12351  df-substr 12352  df-wwlk 30462  df-wwlkn 30463  df-clwwlk 30565  df-clwwlkn 30566
This theorem is referenced by: (None)
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