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Theorem numclwlk1lem2fo 25512
Description: T is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
numclwwlk.t  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
Assertion
Ref Expression
numclwlk1lem2fo  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) -onto-> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Distinct variable groups:    n, E    n, N    n, V    w, C    w, N    C, n, v, w    v, N    n, X, v, w    v, V   
w, E    w, V    w, F    w, G
Allowed substitution hints:    T( w, v, n)    E( v)    F( v, n)    G( v, n)

Proof of Theorem numclwlk1lem2fo
Dummy variables  i  x  p  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 numclwwlk.c . . 3  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
2 numclwwlk.f . . 3  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
3 numclwwlk.g . . 3  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
4 numclwwlk.t . . 3  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
51, 2, 3, 4numclwlk1lem2f 25509 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) --> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
6 elxp 4840 . . . . 5  |-  ( p  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  <->  E. a E. b ( p  = 
<. a ,  b >.  /\  ( a  e.  ( X F ( N  -  2 ) )  /\  b  e.  (
<. V ,  E >. Neighbors  X
) ) ) )
71, 2, 3numclwlk1lem2foa 25508 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) )  ->  ( (
a ++  <" X "> ) ++  <" b "> )  e.  ( X G N ) ) )
87com12 29 . . . . . . . . . 10  |-  ( ( a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) )  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
a ++  <" X "> ) ++  <" b "> )  e.  ( X G N ) ) )
98adantl 464 . . . . . . . . 9  |-  ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N ) ) )
109imp 427 . . . . . . . 8  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N ) )
11 simpl 455 . . . . . . . . 9  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N ) )
12 fveq2 5849 . . . . . . . . . . 11  |-  ( x  =  ( ( a ++ 
<" X "> ) ++  <" b "> )  ->  ( T `  x )  =  ( T `  ( ( a ++  <" X "> ) ++  <" b "> ) ) )
1312eqeq2d 2416 . . . . . . . . . 10  |-  ( x  =  ( ( a ++ 
<" X "> ) ++  <" b "> )  ->  (
p  =  ( T `
 x )  <->  p  =  ( T `  ( ( a ++  <" X "> ) ++  <" b "> ) ) ) )
14 simprr 758 . . . . . . . . . . . 12  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )
151, 2, 3, 4numclwlk1lem2fv 25510 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  ->  ( T `  ( ( a ++  <" X "> ) ++  <" b "> ) )  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) )
1614, 11, 15sylc 59 . . . . . . . . . . 11  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( T `  (
( a ++  <" X "> ) ++  <" b "> ) )  = 
<. ( ( ( a ++ 
<" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. )
1716eqeq2d 2416 . . . . . . . . . 10  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( p  =  ( T `  ( ( a ++  <" X "> ) ++  <" b "> ) )  <->  p  =  <. ( ( ( a ++ 
<" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. ) )
1813, 17sylan9bbr 699 . . . . . . . . 9  |-  ( ( ( ( ( a ++ 
<" X "> ) ++  <" b "> )  e.  ( X G N )  /\  ( ( p  =  <. a ,  b
>.  /\  ( a  e.  ( X F ( N  -  2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) ) ) )  /\  x  =  ( ( a ++ 
<" X "> ) ++  <" b "> ) )  -> 
( p  =  ( T `  x )  <-> 
p  =  <. (
( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) )
19 simprll 764 . . . . . . . . . 10  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  p  =  <. a ,  b >. )
20 nbgraisvtx 24848 . . . . . . . . . . . . . . . . 17  |-  ( V USGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors  X )  ->  b  e.  V ) )
21203ad2ant1 1018 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  ( <. V ,  E >. Neighbors  X )  ->  b  e.  V ) )
22 simp1 997 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  V USGrph  E )
23 uzuzle23 11167 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
24 uznn0sub 11158 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
2523, 24syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e. 
NN0 )
26253ad2ant3 1020 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( N  -  2 )  e. 
NN0 )
27 simp2 998 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  X  e.  V )
281, 2numclwwlkovfel2 25500 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  ( N  -  2 )  e.  NN0  /\  X  e.  V )  ->  (
a  e.  ( X F ( N  - 
2 ) )  <->  ( (
a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 )  /\  ( a `  0
)  =  X ) ) )
2922, 26, 27, 28syl3anc 1230 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( a  e.  ( X F ( N  -  2 ) )  <->  ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 )  /\  ( a `  0
)  =  X ) ) )
30 df-3an 976 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a ) ,  ( a `  0 ) }  e.  ran  E
)  /\  ( # `  a
)  =  ( N  -  2 )  /\  ( a `  0
)  =  X )  <-> 
( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X ) )
3129, 30syl6bb 261 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( a  e.  ( X F ( N  -  2 ) )  <->  ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X ) ) )
32 simplll 760 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  a  e. Word  V )
33 s1cl 12668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( X  e.  V  ->  <" X ">  e. Word  V )
3433adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  <" X ">  e. Word  V )
3534adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  <" X ">  e. Word  V )
3635adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  <" X ">  e. Word  V )
37 s1cl 12668 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( b  e.  V  ->  <" b ">  e. Word  V )
3837adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  <" b ">  e. Word  V )
39 ccatass 12659 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( ( a ++  <" X "> ) ++  <" b "> )  =  ( a ++  ( <" X "> ++  <" b "> ) ) )
4039oveq1d 6293 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( ( ( a ++ 
<" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. )  =  ( ( a ++  ( <" X "> ++  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >. ) )
4132, 36, 38, 40syl3anc 1230 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. )  =  (
( a ++  ( <" X "> ++  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >. ) )
42 ccatcl 12647 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
<" X ">  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( <" X "> ++  <" b "> )  e. Word  V
)
4335, 37, 42syl2an 475 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( <" X "> ++  <" b "> )  e. Word  V )
44 simpr 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
( # `  a )  =  ( N  - 
2 ) )
4544eqcomd 2410 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
( N  -  2 )  =  ( # `  a ) )
4645adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( N  -  2 )  =  ( # `  a
) )
4746adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( N  -  2 )  =  ( # `  a
) )
48 swrdccatid 12778 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e. Word  V  /\  ( <" X "> ++  <" b "> )  e. Word  V  /\  ( N  -  2 )  =  ( # `  a ) )  -> 
( ( a ++  (
<" X "> ++  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >. )  =  a )
4932, 43, 47, 48syl3anc 1230 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( a ++  ( <" X "> ++  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >. )  =  a )
5041, 49eqtr2d 2444 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  a  =  ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) )
51 ovex 6306 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a ++  <" X "> ) ++  <" b "> )  e.  _V
52 lsw 12638 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  _V  ->  ( lastS  `  ( (
a ++  <" X "> ) ++  <" b "> ) )  =  ( ( ( a ++ 
<" X "> ) ++  <" b "> ) `  (
( # `  ( ( a ++  <" X "> ) ++  <" b "> ) )  - 
1 ) ) )
5351, 52ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( lastS  `  (
( a ++  <" X "> ) ++  <" b "> ) )  =  ( ( ( a ++ 
<" X "> ) ++  <" b "> ) `  (
( # `  ( ( a ++  <" X "> ) ++  <" b "> ) )  - 
1 ) )
54 simpl 455 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
a  e. Word  V )
55 ccatcl 12647 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V )  ->  (
a ++  <" X "> )  e. Word  V )
5654, 34, 55syl2an 475 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
a ++  <" X "> )  e. Word  V )
57 lswccats1 12692 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a ++  <" X "> )  e. Word  V  /\  b  e.  V
)  ->  ( lastS  `  (
( a ++  <" X "> ) ++  <" b "> ) )  =  b )
5856, 57sylan 469 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( lastS  `  ( ( a ++  <" X "> ) ++  <" b "> ) )  =  b )
59 ccatlen 12648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( a ++  <" X "> )  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( # `  (
( a ++  <" X "> ) ++  <" b "> ) )  =  ( ( # `  (
a ++  <" X "> ) )  +  (
# `  <" b "> ) ) )
6056, 37, 59syl2an 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 ( ( a ++ 
<" X "> ) ++  <" b "> ) )  =  ( ( # `  (
a ++  <" X "> ) )  +  (
# `  <" b "> ) ) )
6154, 34anim12i 564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
a  e. Word  V  /\  <" X ">  e. Word  V ) )
6261adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
a  e. Word  V  /\  <" X ">  e. Word  V ) )
63 ccatlen 12648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V )  ->  ( # `
 ( a ++  <" X "> )
)  =  ( (
# `  a )  +  ( # `  <" X "> )
) )
6462, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 ( a ++  <" X "> )
)  =  ( (
# `  a )  +  ( # `  <" X "> )
) )
65 s1len 12671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( # `  <" b "> )  =  1
6665a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 <" b "> )  =  1 )
6764, 66oveq12d 6296 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( # `  ( a ++ 
<" X "> ) )  +  (
# `  <" b "> ) )  =  ( ( ( # `  a )  +  (
# `  <" X "> ) )  +  1 ) )
68 s1len 12671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( # `  <" X "> )  =  1
6968a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( # `  <" X "> )  =  1 )
7044, 69oveqan12d 6297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( # `  a )  +  ( # `  <" X "> )
)  =  ( ( N  -  2 )  +  1 ) )
7170oveq1d 6293 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( ( # `  a
)  +  ( # `  <" X "> ) )  +  1 )  =  ( ( ( N  -  2 )  +  1 )  +  1 ) )
72 eluzelcn 11138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  CC )
73 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( N  e.  CC  ->  N  e.  CC )
74 2cnd 10649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( N  e.  CC  ->  2  e.  CC )
7573, 74subcld 9967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( N  e.  CC  ->  ( N  -  2 )  e.  CC )
76 1cnd 9642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( N  e.  CC  ->  1  e.  CC )
7775, 76, 76addassd 9648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( N  e.  CC  ->  (
( ( N  - 
2 )  +  1 )  +  1 )  =  ( ( N  -  2 )  +  ( 1  +  1 ) ) )
78 1p1e2 10690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( 1  +  1 )  =  2
7978a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( N  e.  CC  ->  (
1  +  1 )  =  2 )
8079oveq2d 6294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  ( 1  +  1 ) )  =  ( ( N  -  2 )  +  2 ) )
8177, 80eqtrd 2443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( N  e.  CC  ->  (
( ( N  - 
2 )  +  1 )  +  1 )  =  ( ( N  -  2 )  +  2 ) )
8272, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( (
( N  -  2 )  +  1 )  +  1 )  =  ( ( N  - 
2 )  +  2 ) )
83 2cnd 10649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( N  e.  ( ZZ>= `  3
)  ->  2  e.  CC )
8472, 83npcand 9971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( N  -  2 )  +  2 )  =  N )
8582, 84eqtrd 2443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( (
( N  -  2 )  +  1 )  +  1 )  =  N )
8685adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( ( N  -  2 )  +  1 )  +  1 )  =  N )
8786adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( ( N  - 
2 )  +  1 )  +  1 )  =  N )
8871, 87eqtrd 2443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( ( # `  a
)  +  ( # `  <" X "> ) )  +  1 )  =  N )
8988adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( ( # `  a
)  +  ( # `  <" X "> ) )  +  1 )  =  N )
9060, 67, 893eqtrd 2447 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 ( ( a ++ 
<" X "> ) ++  <" b "> ) )  =  N )
9190oveq1d 6293 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( # `  ( ( a ++  <" X "> ) ++  <" b "> ) )  - 
1 )  =  ( N  -  1 ) )
9291fveq2d 5853 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( ( a ++  <" X "> ) ++  <" b "> ) `  ( ( # `
 ( ( a ++ 
<" X "> ) ++  <" b "> ) )  - 
1 ) )  =  ( ( ( a ++ 
<" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) )
9353, 58, 923eqtr3a 2467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  b  =  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) )
9450, 93opeq12d 4167 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. )
9594exp31 602 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
( ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
96953ad2antl1 1159 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a ) ,  ( a `  0 ) }  e.  ran  E
)  /\  ( # `  a
)  =  ( N  -  2 ) )  ->  ( ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
9796adantr 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X )  ->  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( b  e.  V  -> 
<. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. ) ) )
9897com12 29 . . . . . . . . . . . . . . . . . . 19  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X )  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
99983adant1 1015 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X )  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
10031, 99sylbid 215 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( a  e.  ( X F ( N  -  2 ) )  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
101100com23 78 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  V  ->  ( a  e.  ( X F ( N  -  2 ) )  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
10221, 101syld 42 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  ( <. V ,  E >. Neighbors  X )  ->  (
a  e.  ( X F ( N  - 
2 ) )  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. ) ) )
103102com13 80 . . . . . . . . . . . . . 14  |-  ( a  e.  ( X F ( N  -  2 ) )  ->  (
b  e.  ( <. V ,  E >. Neighbors  X
)  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
104103imp 427 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) )  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. ) )
105104adantl 464 . . . . . . . . . . . 12  |-  ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. ) )
106105imp 427 . . . . . . . . . . 11  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) ) >. )
107106adantl 464 . . . . . . . . . 10  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  <. a ,  b >.  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. )
10819, 107eqtrd 2443 . . . . . . . . 9  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  p  =  <. ( ( ( a ++  <" X "> ) ++  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a ++  <" X "> ) ++  <" b "> ) `  ( N  -  1 ) )
>. )
10911, 18, 108rspcedvd 3165 . . . . . . . 8  |-  ( ( ( ( a ++  <" X "> ) ++  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) )
11010, 109mpancom 667 . . . . . . 7  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) )
111110ex 432 . . . . . 6  |-  ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
112111exlimivv 1744 . . . . 5  |-  ( E. a E. b ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
1136, 112sylbi 195 . . . 4  |-  ( p  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  -> 
( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  ->  E. x  e.  ( X G N ) p  =  ( T `
 x ) ) )
114113impcom 428 . . 3  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  p  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )  ->  E. x  e.  ( X G N ) p  =  ( T `
 x ) )
115114ralrimiva 2818 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  A. p  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) E. x  e.  ( X G N ) p  =  ( T `  x ) )
116 dffo3 6024 . 2  |-  ( T : ( X G N ) -onto-> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  <->  ( T : ( X G N ) --> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  /\  A. p  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
1175, 115, 116sylanbrc 662 1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) -onto-> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842   A.wral 2754   E.wrex 2755   {crab 2758   _Vcvv 3059   {cpr 3974   <.cop 3978   class class class wbr 4395    |-> cmpt 4453    X. cxp 4821   ran crn 4824   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525    - cmin 9841   2c2 10626   3c3 10627   NN0cn0 10836   ZZ>=cuz 11127  ..^cfzo 11854   #chash 12452  Word cword 12583   lastS clsw 12584   ++ cconcat 12585   <"cs1 12586   substr csubstr 12587   USGrph cusg 24747   Neighbors cnbgra 24834   ClWWalksN cclwwlkn 25166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-lsw 12592  df-concat 12593  df-s1 12594  df-substr 12595  df-usgra 24750  df-nbgra 24837  df-clwwlk 25168  df-clwwlkn 25169
This theorem is referenced by:  numclwlk1lem2f1o  25513
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