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Theorem rusgranumwlkl1 30484
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 30473 . . . . . . . . . . 11  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
2 usgrav 23205 . . . . . . . . . . 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 10591 . . . . . . . . . . 11  |-  1  e.  NN0
54a1i 11 . . . . . . . . . 10  |-  ( P  e.  V  ->  1  e.  NN0 )
63, 5anim12i 563 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  1  e.  NN0 ) )
7 df-3an 962 . . . . . . . . 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 30243 . . . . . . . . 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 30242 . . . . . . . . . . 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 1003 . . . . . . . . . 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 699 . . . . . . . . 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 699 . . . . . 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 10431 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
1716eqeq2i 2451 . . . . . . . . . . 11  |-  ( (
# `  w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 )
1817a1i 11 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  ->  ( ( # `
 w )  =  ( 1  +  1 )  <->  ( # `  w
)  =  2 ) )
1918anbi2d 698 . . . . . . . . 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 964 . . . . . . . . . . . . 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 5688 . . . . . . . . . . . . . . . . 17  |-  ( w  =  (/)  ->  ( # `  w )  =  (
# `  (/) ) )
23 hash0 12131 . . . . . . . . . . . . . . . . 17  |-  ( # `  (/) )  =  0
2422, 23syl6eq 2489 . . . . . . . . . . . . . . . 16  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
25 2ne0 10410 . . . . . . . . . . . . . . . . . 18  |-  2  =/=  0
2625nesymi 2646 . . . . . . . . . . . . . . . . 17  |-  -.  0  =  2
27 eqeq1 2447 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  0  ->  (
( # `  w )  =  2  <->  0  = 
2 ) )
2826, 27mtbiri 303 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  0  ->  -.  ( # `  w )  =  2 )
2924, 28syl 16 . . . . . . . . . . . . . . 15  |-  ( w  =  (/)  ->  -.  ( # `
 w )  =  2 )
3029necon2ai 2654 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  2  ->  w  =/=  (/) )
3130adantl 463 . . . . . . . . . . . . 13  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  w  =/=  (/) )
3231biantrurd 505 . . . . . . . . . . . 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 6097 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  ( 2  -  1 ) )
34 2m1e1 10432 . . . . . . . . . . . . . . . . . 18  |-  ( 2  -  1 )  =  1
3533, 34syl6eq 2489 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  2  ->  (
( # `  w )  -  1 )  =  1 )
3635oveq2d 6106 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  2  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 1 ) )
3736adantl 463 . . . . . . . . . . . . . . 15  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  P  e.  V
)  /\  ( # `  w
)  =  2 )  ->  ( 0..^ ( ( # `  w
)  -  1 ) )  =  ( 0..^ 1 ) )
3837raleqdv 2921 . . . . . . . . . . . . . 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 11608 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 1 )  =  {
0 }
4039raleqi 2919 . . . . . . . . . . . . . . 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 9376 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
42 fveq2 5688 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
w `  i )  =  ( w ` 
0 ) )
43 oveq1 6097 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
44 0p1e1 10429 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  +  1 )  =  1
4543, 44syl6eq 2489 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  0  ->  (
i  +  1 )  =  1 )
4645fveq2d 5692 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  0  ->  (
w `  ( i  +  1 ) )  =  ( w ` 
1 ) )
4742, 46preq12d 3959 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  0  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w ` 
0 ) ,  ( w `  1 ) } )
4847eleq1d 2507 . . . . . . . . . . . . . . . . 17  |-  ( i  =  0  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  0
) ,  ( w `
 1 ) }  e.  ran  E ) )
4948ralsng 3909 . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . 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 635 . . . . . . . . 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 699 . . . . . . 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 644 . . . . . . 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 644 . . . . . . . 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 448 . . . . . . . . . 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 962 . . . . . . . . . 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 689 . . . . . . . 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 2555 . . . 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 2722 . . . 4  |-  { w  e.  ( ( V WWalksN  E
) `  1 )  |  ( w ` 
0 )  =  P }  =  { w  |  ( w  e.  ( ( V WWalksN  E
) `  1 )  /\  ( w `  0
)  =  P ) }
71 df-rab 2722 . . . 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 2498 . . 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 5692 . 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 30483 . 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 2473 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 960    = wceq 1364    e. wcel 1761   {cab 2427    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970   (/)c0 3634   {csn 3874   {cpr 3876   <.cop 3880   class class class wbr 4289   ran crn 4837   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281    - cmin 9591   2c2 10367   NN0cn0 10575  ..^cfzo 11544   #chash 12099  Word cword 12217   USGrph cusg 23199   WWalks cwwlk 30236   WWalksN cwwlkn 30237   RegUSGrph crusgra 30465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-xadd 11086  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-usgra 23201  df-nbgra 23267  df-vdgr 23499  df-wwlk 30238  df-wwlkn 30239  df-rgra 30466  df-rusgra 30467
This theorem is referenced by:  rusgranumwlkb1  30497
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