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Theorem wwlkexthasheq 30537
Description: The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
Assertion
Ref Expression
wwlkexthasheq  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )  =  ( # `  {
n  e.  V  |  { ( lastS  `  W ) ,  n }  e.  ran  E } ) )
Distinct variable groups:    n, E, w    n, N, w    n, V, w    n, W, w

Proof of Theorem wwlkexthasheq
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fvex 5812 . . . 4  |-  ( ( V WWalksN  E ) `  ( N  +  1 ) )  e.  _V
2 rabexg 4553 . . . 4  |-  ( ( ( V WWalksN  E ) `  ( N  +  1 ) )  e.  _V  ->  { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }  e.  _V )
31, 2mp1i 12 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  e.  _V )
4 wwlknprop 30491 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
5 simpll 753 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) )  ->  V  e.  _V )
6 rabexg 4553 . . . 4  |-  ( V  e.  _V  ->  { n  e.  V  |  {
( lastS  `  W ) ,  n }  e.  ran  E }  e.  _V )
74, 5, 63syl 20 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { n  e.  V  |  {
( lastS  `  W ) ,  n }  e.  ran  E }  e.  _V )
83, 7jca 532 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  e.  _V  /\ 
{ n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }  e.  _V ) )
9 wwlkextbij 30536 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  E. f 
f : { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E } )
10 hasheqf1oi 12243 . 2  |-  ( ( { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }  e.  _V  /\  { n  e.  V  |  { ( lastS  `  W ) ,  n }  e.  ran  E }  e.  _V )  ->  ( E. f  f : { w  e.  (
( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } -1-1-onto-> { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }  ->  (
# `  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) } )  =  ( # `  {
n  e.  V  |  { ( lastS  `  W ) ,  n }  e.  ran  E } ) ) )
118, 9, 10sylc 60 1  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )  =  ( # `  {
n  e.  V  |  { ( lastS  `  W ) ,  n }  e.  ran  E } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   {crab 2803   _Vcvv 3078   {cpr 3990   <.cop 3994   ran crn 4952   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   0cc0 9397   1c1 9398    + caddc 9400   NN0cn0 10694   #chash 12224  Word cword 12343   lastS clsw 12344   substr csubstr 12347   WWalksN cwwlkn 30483
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-lsw 12352  df-concat 12353  df-s1 12354  df-substr 12355  df-wwlk 30484  df-wwlkn 30485
This theorem is referenced by:  rusgranumwlks  30745
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