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Theorem rusgranumwlkl1 24623
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 24607 . . . . . . . . . . 11  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 usgrav 24014 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
31, 2syl 16 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V  e.  _V  /\  E  e.  _V )
)
4 1nn0 10807 . . . . . . . . . . 11  |-  1  e.  NN0
54a1i 11 . . . . . . . . . 10  |-  ( P  e.  V  ->  1  e.  NN0 )
63, 5anim12i 566 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
7 df-3an 975 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  1  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
86, 7sylibr 212 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  1  e.  NN0 ) )
9 iswwlkn 24360 . . . . . . . . 9  |-  ( ( 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 24359 . . . . . . . . . . 11  |-  ( ( 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 . . . . . . . . . 10  |-  ( ( 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 . . . . . . . . 9  |-  ( ( 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 . . . . . . . 8  |-  ( ( 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 . . . . . . 7  |-  ( (
<. 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 . . . . . 6  |-  ( (
<. 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 10645 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
1716eqeq2i 2485 . . . . . . . . . . 11  |-  ( (
# `  w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 )
1817a1i 11 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( # `
 w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 ) )
1918anbi2d 703 . . . . . . . . 9  |-  ( (
<. 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 . . . . . . . . . . . . 13  |-  ( ( 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 . . . . . . . . . . . 12  |-  ( ( ( <. 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 5864 . . . . . . . . . . . . . . . . 17  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
23 hash0 12401 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
2422, 23syl6eq 2524 . . . . . . . . . . . . . . . 16  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
25 2ne0 10624 . . . . . . . . . . . . . . . . . 18  |-  2  =/=  0
2625nesymi 2740 . . . . . . . . . . . . . . . . 17  |-  -.  0  =  2
27 eqeq1 2471 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  0  ->  (
( # `  w )  =  2  <->  0  = 
2 ) )
2826, 27mtbiri 303 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  0  ->  -.  ( # `  w )  =  2 )
2924, 28syl 16 . . . . . . . . . . . . . . 15  |-  ( w  =  (/)  ->  -.  ( # `
 w )  =  2 )
3029necon2ai 2702 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  2  ->  w  =/=  (/) )
3130adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  w  =/=  (/) )
3231biantrurd 508 . . . . . . . . . . . 12  |-  ( ( ( <. 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 6289 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  ( 2  -  1 ) )
34 2m1e1 10646 . . . . . . . . . . . . . . . . . 18  |-  ( 2  -  1 )  =  1
3533, 34syl6eq 2524 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  1 )
3635oveq2d 6298 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  2  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 1 ) )
3736adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ 1 ) )
3837raleqdv 3064 . . . . . . . . . . . . . 14  |-  ( ( ( <. 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 11861 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 1 )  =  {
0 }
4039raleqi 3062 . . . . . . . . . . . . . . 15  |-  ( 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 9586 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
42 fveq2 5864 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
w `  i )  =  ( w ` 
0 ) )
43 oveq1 6289 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
44 0p1e1 10643 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  +  1 )  =  1
4543, 44syl6eq 2524 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  0  ->  (
i  +  1 )  =  1 )
4645fveq2d 5868 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
w `  ( i  +  1 ) )  =  ( w ` 
1 ) )
4742, 46preq12d 4114 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  0  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w ` 
0 ) ,  ( w `  1 ) } )
4847eleq1d 2536 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
4948ralsng 4062 . . . . . . . . . . . . . . . 16  |-  ( 0  e.  _V  ->  ( A. i  e.  { 0 }  { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E  <->  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
) )
5041, 49ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( A. i  e.  { 0 }  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )
5140, 50bitri 249 . . . . . . . . . . . . . 14  |-  ( A. i  e.  ( 0..^ 1 ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)
5238, 51syl6bb 261 . . . . . . . . . . . . 13  |-  ( ( ( <. 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 ) )
5352anbi2d 703 . . . . . . . . . . . 12  |-  ( ( ( <. 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 ) ) )
5421, 32, 533bitr2d 281 . . . . . . . . . . 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  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) )
5554ex 434 . . . . . . . . . 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 ) ) ) )
5655pm5.32rd 640 . . . . . . . . 9  |-  ( (
<. 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 ) ) )
5719, 56bitrd 253 . . . . . . . 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  e. Word  V  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E )  /\  ( # `  w )  =  2 ) ) )
5857anbi1d 704 . . . . . . 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 `  0 )  =  P )  <->  ( (
( w  e. Word  V  /\  { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E
)  /\  ( # `  w
)  =  2 )  /\  ( w ` 
0 )  =  P ) ) )
59 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 ) ) )
6058, 59syl6bb 261 . . . . . 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 ) ) ) )
61 anass 649 . . . . . . . 8  |-  ( ( ( 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 ) ) ) )
62 ancom 450 . . . . . . . . . 10  |-  ( ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  ran  E  /\  ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P ) )  <->  ( ( (
# `  w )  =  2  /\  (
w `  0 )  =  P )  /\  {
( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
63 df-3an 975 . . . . . . . . . 10  |-  ( ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E )  <-> 
( ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P )  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) )
6462, 63bitr4i 252 . . . . . . . . 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 ) )
6564anbi2i 694 . . . . . . . 8  |-  ( ( 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 ) ) )
6661, 65bitri 249 . . . . . . 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 ) ) )
6766a1i 11 . . . . . 6  |-  ( (
<. 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 ) ) ) )
6815, 60, 673bitrd 279 . . . . 5  |-  ( (
<. 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 ) ) ) )
6968abbidv 2603 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  { w  |  ( w  e.  ( ( V WWalksN  E
) `  1 )  /\  ( w `  0
)  =  P ) }  =  { w  |  ( w  e. Word  V  /\  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  ran  E ) ) } )
70 df-rab 2823 . . . 4  |-  { w  e.  ( ( V WWalksN  E
) `  1 )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  1 )  /\  ( w `  0
)  =  P ) }
71 df-rab 2823 . . . 4  |-  { w  e. Word  V  |  ( (
# `  w )  =  2  /\  (
w `  0 )  =  P  /\  { ( w `  0 ) ,  ( w ` 
1 ) }  e.  ran  E ) }  =  { w  |  (
w  e. Word  V  /\  ( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) ) }
7269, 70, 713eqtr4g 2533 . . 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 ) } )
7372fveq2d 5868 . 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 ) } ) )
74 rusgranumwwlkl1 24622 . 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 )
7573, 74eqtrd 2508 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 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   (/)c0 3785   {csn 4027   {cpr 4029   <.cop 4033   class class class wbr 4447   ran crn 5000   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801   2c2 10581   NN0cn0 10791  ..^cfzo 11788   #chash 12369  Word cword 12496   USGrph cusg 24006   WWalks cwwlk 24353   WWalksN cwwlkn 24354   RegUSGrph crusgra 24599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-xadd 11315  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-usgra 24009  df-nbgra 24096  df-wwlk 24355  df-wwlkn 24356  df-vdgr 24570  df-rgra 24600  df-rusgra 24601
This theorem is referenced by:  rusgranumwlkb1  24630
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