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Theorem numclwwlkovf2num 30821
Description: In a k regular graph, therere are k closed walks of length 2 starting at a fixed vertex. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
Assertion
Ref Expression
numclwwlkovf2num  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  X  e.  V
)  ->  ( # `  ( X F 2 ) )  =  K )
Distinct variable groups:    n, E    n, V    w, C, n, v    n, X, v, w    v, V    w, E    w, V
Allowed substitution hints:    E( v)    F( w, v, n)    K( w, v, n)

Proof of Theorem numclwwlkovf2num
StepHypRef Expression
1 rusisusgra 30691 . . . 4  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
3 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
42, 3numclwwlkovf2 30820 . . . 4  |-  ( ( V USGrph  E  /\  X  e.  V )  ->  ( X F 2 )  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E  /\  ( w ` 
0 )  =  X ) } )
51, 4sylan 471 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  X  e.  V
)  ->  ( X F 2 )  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E  /\  ( w ` 
0 )  =  X ) } )
65fveq2d 5798 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  X  e.  V
)  ->  ( # `  ( X F 2 ) )  =  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) } ) )
7 3ancomb 974 . . . . . 6  |-  ( ( ( # `  w
)  =  2  /\  ( w `  0
)  =  X  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  <-> 
( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) )
87a1i 11 . . . . 5  |-  ( w  e. Word  V  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  X  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  <-> 
( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) ) )
98rabbiia 3061 . . . 4  |-  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  X  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) }  =  { w  e. Word  V  | 
( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) }
109fveq2i 5797 . . 3  |-  ( # `  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  X  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) } )  =  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) } )
11 rusgranumwlkl1lem1 30701 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  X  e.  V
)  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  X  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } )  =  K )
1210, 11syl5eqr 2507 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  X  e.  V
)  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\ 
{ ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( w `  0
)  =  X ) } )  =  K )
136, 12eqtrd 2493 1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  X  e.  V
)  ->  ( # `  ( X F 2 ) )  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2800   {cpr 3982   <.cop 3986   class class class wbr 4395    |-> cmpt 4453   ran crn 4944   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   0cc0 9388   1c1 9389   2c2 10477   NN0cn0 10685   #chash 12215  Word cword 12334   USGrph cusg 23411   ClWWalksN cclwwlkn 30557   RegUSGrph crusgra 30683
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-xadd 11196  df-fz 11550  df-fzo 11661  df-hash 12216  df-word 12342  df-lsw 12343  df-usgra 23413  df-nbgra 23479  df-vdgr 23711  df-clwwlk 30559  df-clwwlkn 30560  df-rgra 30684  df-rusgra 30685
This theorem is referenced by:  numclwwlk5lem  30847
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