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Theorem rusgranumwlkl1 25674
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 25658 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 usgrav 25064 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
31, 2syl 17 . . . . . . . . 9  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V  e.  _V  /\  E  e.  _V )
)
4 1nn0 10893 . . . . . . . . . 10  |-  1  e.  NN0
54a1i 11 . . . . . . . . 9  |-  ( P  e.  V  ->  1  e.  NN0 )
63, 5anim12i 568 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
7 df-3an 984 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
86, 7sylibr 215 . . . . . . 7  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  1  e.  NN0 ) )
9 iswwlkn 25411 . . . . . . . 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 25410 . . . . . . . . . 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 1025 . . . . . . . . 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 709 . . . . . . . 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 256 . . . . . . 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 17 . . . . . 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 709 . . . . 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 10731 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
1716eqeq2i 2440 . . . . . . . . . 10  |-  ( (
# `  w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 )
1817a1i 11 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( # `
 w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 ) )
1918anbi2d 708 . . . . . . . 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 986 . . . . . . . . . . . 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 5882 . . . . . . . . . . . . . . . 16  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
23 hash0 12555 . . . . . . . . . . . . . . . 16  |-  ( # `  (/) )  =  0
2422, 23syl6eq 2479 . . . . . . . . . . . . . . 15  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
25 2ne0 10710 . . . . . . . . . . . . . . . . 17  |-  2  =/=  0
2625nesymi 2693 . . . . . . . . . . . . . . . 16  |-  -.  0  =  2
27 eqeq1 2426 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  0  ->  (
( # `  w )  =  2  <->  0  = 
2 ) )
2826, 27mtbiri 304 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  0  ->  -.  ( # `  w )  =  2 )
2924, 28syl 17 . . . . . . . . . . . . . 14  |-  ( w  =  (/)  ->  -.  ( # `
 w )  =  2 )
3029necon2ai 2655 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  2  ->  w  =/=  (/) )
3130adantl 467 . . . . . . . . . . . 12  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  w  =/=  (/) )
3231biantrurd 510 . . . . . . . . . . 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 6313 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  ( 2  -  1 ) )
34 2m1e1 10732 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
3533, 34syl6eq 2479 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  1 )
3635oveq2d 6322 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  2  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 1 ) )
3736adantl 467 . . . . . . . . . . . . . 14  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ 1 ) )
3837raleqdv 3028 . . . . . . . . . . . . 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 12002 . . . . . . . . . . . . . . 15  |-  ( 0..^ 1 )  =  {
0 }
4039raleqi 3026 . . . . . . . . . . . . . 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 9645 . . . . . . . . . . . . . . 15  |-  0  e.  _V
42 fveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  (
w `  i )  =  ( w ` 
0 ) )
43 oveq1 6313 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
44 0p1e1 10729 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  +  1 )  =  1
4543, 44syl6eq 2479 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  0  ->  (
i  +  1 )  =  1 )
4645fveq2d 5886 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  (
w `  ( i  +  1 ) )  =  ( w ` 
1 ) )
4742, 46preq12d 4087 . . . . . . . . . . . . . . . 16  |-  ( i  =  0  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w ` 
0 ) ,  ( w `  1 ) } )
4847eleq1d 2491 . . . . . . . . . . . . . . 15  |-  ( i  =  0  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
4941, 48ralsn 4038 . . . . . . . . . . . . . 14  |-  ( A. i  e.  { 0 }  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )
5040, 49bitri 252 . . . . . . . . . . . . 13  |-  ( A. i  e.  ( 0..^ 1 ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)
5138, 50syl6bb 264 . . . . . . . . . . . 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 708 . . . . . . . . . . 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 284 . . . . . . . . . 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 435 . . . . . . . . 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 644 . . . . . . . 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 256 . . . . . . 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 709 . . . . . 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 653 . . . . . 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 264 . . . . 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 653 . . . . . . 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 451 . . . . . . . . 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 984 . . . . . . . . 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 255 . . . . . . . 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 698 . . . . . . 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 252 . . . . . 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 282 . . . 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 3069 . . 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 5886 . 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 25673 . 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 2463 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   {crab 2775   _Vcvv 3080   (/)c0 3761   {csn 3998   {cpr 4000   <.cop 4004   class class class wbr 4423   ran crn 4854   ` cfv 5601  (class class class)co 6306   0cc0 9547   1c1 9548    + caddc 9550    - cmin 9868   2c2 10667   NN0cn0 10877  ..^cfzo 11923   #chash 12522  Word cword 12661   USGrph cusg 25056   WWalks cwwlk 25404   WWalksN cwwlkn 25405   RegUSGrph crusgra 25650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-card 8382  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-n0 10878  df-z 10946  df-uz 11168  df-xadd 11418  df-fz 11793  df-fzo 11924  df-hash 12523  df-word 12669  df-usgra 25059  df-nbgra 25147  df-wwlk 25406  df-wwlkn 25407  df-vdgr 25621  df-rgra 25651  df-rusgra 25652
This theorem is referenced by:  rusgranumwlkb1  25681
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