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Theorem rusgranumwlkl1 25073
Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
Assertion
Ref Expression
rusgranumwlkl1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  1
)  |  ( w `
 0 )  =  P } )  =  K )
Distinct variable groups:    w, E    w, K    w, P    w, V

Proof of Theorem rusgranumwlkl1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 rusisusgra 25057 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 usgrav 24464 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
31, 2syl 16 . . . . . . . . 9  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V  e.  _V  /\  E  e.  _V )
)
4 1nn0 10832 . . . . . . . . . 10  |-  1  e.  NN0
54a1i 11 . . . . . . . . 9  |-  ( P  e.  V  ->  1  e.  NN0 )
63, 5anim12i 566 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
7 df-3an 975 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
86, 7sylibr 212 . . . . . . 7  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  1  e.  NN0 ) )
9 iswwlkn 24810 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  ->  (
w  e.  ( ( V WWalksN  E ) `  1
)  <->  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( 1  +  1 ) ) ) )
10 iswwlk 24809 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
11103adant3 1016 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  ->  (
w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
1211anbi1d 704 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  ->  (
( w  e.  ( V WWalks  E )  /\  ( # `  w )  =  ( 1  +  1 ) )  <->  ( (
w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( 1  +  1 ) ) ) )
139, 12bitrd 253 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  ->  (
w  e.  ( ( V WWalksN  E ) `  1
)  <->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( 1  +  1 ) ) ) )
148, 13syl 16 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( w  e.  ( ( V WWalksN  E
) `  1 )  <->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( 1  +  1 ) ) ) )
1514anbi1d 704 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
w  e.  ( ( V WWalksN  E ) `  1
)  /\  ( w `  0 )  =  P )  <->  ( (
( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( 1  +  1 ) )  /\  ( w ` 
0 )  =  P ) ) )
16 1p1e2 10670 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
1716eqeq2i 2475 . . . . . . . . . 10  |-  ( (
# `  w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 )
1817a1i 11 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( # `
 w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 ) )
1918anbi2d 703 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( 1  +  1 ) )  <-> 
( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E )  /\  ( # `
 w )  =  2 ) ) )
20 3anass 977 . . . . . . . . . . . 12  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( w  =/=  (/)  /\  ( w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
2120a1i 11 . . . . . . . . . . 11  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( w  =/=  (/)  /\  ( w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) ) )
22 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
23 hash0 12439 . . . . . . . . . . . . . . . 16  |-  ( # `  (/) )  =  0
2422, 23syl6eq 2514 . . . . . . . . . . . . . . 15  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
25 2ne0 10649 . . . . . . . . . . . . . . . . 17  |-  2  =/=  0
2625nesymi 2730 . . . . . . . . . . . . . . . 16  |-  -.  0  =  2
27 eqeq1 2461 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  0  ->  (
( # `  w )  =  2  <->  0  = 
2 ) )
2826, 27mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  0  ->  -.  ( # `  w )  =  2 )
2924, 28syl 16 . . . . . . . . . . . . . 14  |-  ( w  =  (/)  ->  -.  ( # `
 w )  =  2 )
3029necon2ai 2692 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  2  ->  w  =/=  (/) )
3130adantl 466 . . . . . . . . . . . 12  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  w  =/=  (/) )
3231biantrurd 508 . . . . . . . . . . 11  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( ( w  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  <->  ( w  =/=  (/)  /\  ( w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) ) )
33 oveq1 6303 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  ( 2  -  1 ) )
34 2m1e1 10671 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
3533, 34syl6eq 2514 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  1 )
3635oveq2d 6312 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  2  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 1 ) )
3736adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ 1 ) )
3837raleqdv 3060 . . . . . . . . . . . . 13  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ 1 ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E ) )
39 fzo01 11899 . . . . . . . . . . . . . . 15  |-  ( 0..^ 1 )  =  {
0 }
4039raleqi 3058 . . . . . . . . . . . . . 14  |-  ( A. i  e.  ( 0..^ 1 ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  {
0 }  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E )
41 c0ex 9607 . . . . . . . . . . . . . . 15  |-  0  e.  _V
42 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  (
w `  i )  =  ( w ` 
0 ) )
43 oveq1 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
44 0p1e1 10668 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  +  1 )  =  1
4543, 44syl6eq 2514 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  0  ->  (
i  +  1 )  =  1 )
4645fveq2d 5876 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  (
w `  ( i  +  1 ) )  =  ( w ` 
1 ) )
4742, 46preq12d 4119 . . . . . . . . . . . . . . . 16  |-  ( i  =  0  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w ` 
0 ) ,  ( w `  1 ) } )
4847eleq1d 2526 . . . . . . . . . . . . . . 15  |-  ( i  =  0  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
4941, 48ralsn 4071 . . . . . . . . . . . . . 14  |-  ( A. i  e.  { 0 }  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )
5040, 49bitri 249 . . . . . . . . . . . . 13  |-  ( A. i  e.  ( 0..^ 1 ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)
5138, 50syl6bb 261 . . . . . . . . . . . 12  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
5251anbi2d 703 . . . . . . . . . . 11  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( ( w  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  <->  ( w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) )
5321, 32, 523bitr2d 281 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) )
5453ex 434 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( # `
 w )  =  2  ->  ( (
w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) ) )
5554pm5.32rd 640 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  2 )  <-> 
( ( w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E )  /\  ( # `  w )  =  2 ) ) )
5619, 55bitrd 253 . . . . . . 7  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( 1  +  1 ) )  <-> 
( ( w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E )  /\  ( # `  w )  =  2 ) ) )
5756anbi1d 704 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E )  /\  ( # `
 w )  =  ( 1  +  1 ) )  /\  (
w `  0 )  =  P )  <->  ( (
( w  e. Word  V  /\  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)  /\  ( # `  w
)  =  2 )  /\  ( w ` 
0 )  =  P ) ) )
58 anass 649 . . . . . 6  |-  ( ( ( ( w  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E )  /\  ( # `  w )  =  2 )  /\  ( w `
 0 )  =  P )  <->  ( (
w  e. Word  V  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  /\  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P ) ) )
5957, 58syl6bb 261 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E )  /\  ( # `
 w )  =  ( 1  +  1 ) )  /\  (
w `  0 )  =  P )  <->  ( (
w  e. Word  V  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  /\  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P ) ) ) )
60 anass 649 . . . . . . 7  |-  ( ( ( w  e. Word  V  /\  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)  /\  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P ) )  <->  ( w  e. Word  V  /\  ( { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E  /\  ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P ) ) ) )
61 ancom 450 . . . . . . . . 9  |-  ( ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P ) )  <->  ( ( (
# `  w )  =  2  /\  (
w `  0 )  =  P )  /\  {
( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
62 df-3an 975 . . . . . . . . 9  |-  ( ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  <-> 
( ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P )  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) )
6361, 62bitr4i 252 . . . . . . . 8  |-  ( ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P ) )  <->  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) )
6463anbi2i 694 . . . . . . 7  |-  ( ( w  e. Word  V  /\  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P ) ) )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) ) )
6560, 64bitri 249 . . . . . 6  |-  ( ( ( w  e. Word  V  /\  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)  /\  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P ) )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) ) )
6665a1i 11 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
( w  e. Word  V  /\  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)  /\  ( ( # `
 w )  =  2  /\  ( w `
 0 )  =  P ) )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) ) ) )
6715, 59, 663bitrd 279 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( (
w  e.  ( ( V WWalksN  E ) `  1
)  /\  ( w `  0 )  =  P )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) ) ) )
6867rabbidva2 3099 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  { w  e.  ( ( V WWalksN  E
) `  1 )  |  ( w ` 
0 )  =  P }  =  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) } )
6968fveq2d 5876 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  1
)  |  ( w `
 0 )  =  P } )  =  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } ) )
70 rusgranumwwlkl1 25072 . 2  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e. Word  V  | 
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) } )  =  K )
7169, 70eqtrd 2498 1  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( # `  {
w  e.  ( ( V WWalksN  E ) `  1
)  |  ( w `
 0 )  =  P } )  =  K )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   (/)c0 3793   {csn 4032   {cpr 4034   <.cop 4038   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   2c2 10606   NN0cn0 10816  ..^cfzo 11820   #chash 12407  Word cword 12537   USGrph cusg 24456   WWalks cwwlk 24803   WWalksN cwwlkn 24804   RegUSGrph crusgra 25049
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
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-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  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-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-xadd 11344  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-usgra 24459  df-nbgra 24546  df-wwlk 24805  df-wwlkn 24806  df-vdgr 25020  df-rgra 25050  df-rusgra 25051
This theorem is referenced by:  rusgranumwlkb1  25080
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