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Theorem rusgranumwlkl1 30702
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 30691 . . . . . . . . . . 11  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 usgrav 23417 . . . . . . . . . . 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 10701 . . . . . . . . . . 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 967 . . . . . . . . 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 30461 . . . . . . . . 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 30460 . . . . . . . . . . 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 1008 . . . . . . . . . 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 10541 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
1716eqeq2i 2470 . . . . . . . . . . 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 969 . . . . . . . . . . . . 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 5794 . . . . . . . . . . . . . . . . 17  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
23 hash0 12247 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
2422, 23syl6eq 2509 . . . . . . . . . . . . . . . 16  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
25 2ne0 10520 . . . . . . . . . . . . . . . . . 18  |-  2  =/=  0
2625nesymi 2722 . . . . . . . . . . . . . . . . 17  |-  -.  0  =  2
27 eqeq1 2456 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  0  ->  (
( # `  w )  =  2  <->  0  = 
2 ) )
2826, 27mtbiri 303 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  0  ->  -.  ( # `  w )  =  2 )
2924, 28syl 16 . . . . . . . . . . . . . . 15  |-  ( w  =  (/)  ->  -.  ( # `
 w )  =  2 )
3029necon2ai 2684 . . . . . . . . . . . . . 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 6202 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  ( 2  -  1 ) )
34 2m1e1 10542 . . . . . . . . . . . . . . . . . 18  |-  ( 2  -  1 )  =  1
3533, 34syl6eq 2509 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  1 )
3635oveq2d 6211 . . . . . . . . . . . . . . . 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 3023 . . . . . . . . . . . . . 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 11724 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 1 )  =  {
0 }
4039raleqi 3021 . . . . . . . . . . . . . . 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 9486 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
42 fveq2 5794 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
w `  i )  =  ( w ` 
0 ) )
43 oveq1 6202 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
44 0p1e1 10539 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  +  1 )  =  1
4543, 44syl6eq 2509 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  0  ->  (
i  +  1 )  =  1 )
4645fveq2d 5798 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
w `  ( i  +  1 ) )  =  ( w ` 
1 ) )
4742, 46preq12d 4065 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  0  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w ` 
0 ) ,  ( w `  1 ) } )
4847eleq1d 2521 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
4948ralsng 4015 . . . . . . . . . . . . . . . 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 967 . . . . . . . . . 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 2588 . . . 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 2805 . . . 4  |-  { w  e.  ( ( V WWalksN  E
) `  1 )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  1 )  /\  ( w `  0
)  =  P ) }
71 df-rab 2805 . . . 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 2518 . . 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 5798 . 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 rusgranumwlkl1lem1 30701 . 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 2493 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 965    = wceq 1370    e. wcel 1758   {cab 2437    =/= wne 2645   A.wral 2796   {crab 2800   _Vcvv 3072   (/)c0 3740   {csn 3980   {cpr 3982   <.cop 3986   class class class wbr 4395   ran crn 4944   ` cfv 5521  (class class class)co 6195   0cc0 9388   1c1 9389    + caddc 9391    - cmin 9701   2c2 10477   NN0cn0 10685  ..^cfzo 11660   #chash 12215  Word cword 12334   USGrph cusg 23411   WWalks cwwlk 30454   WWalksN cwwlkn 30455   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-usgra 23413  df-nbgra 23479  df-vdgr 23711  df-wwlk 30456  df-wwlkn 30457  df-rgra 30684  df-rusgra 30685
This theorem is referenced by:  rusgranumwlkb1  30715
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