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Theorem usg2wlkeq 30413
Description: Conditions for two walks within the same undirected simple graph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.)
Assertion
Ref Expression
usg2wlkeq  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) ) ) )
Distinct variable groups:    y, A    y, B    y, N    y, E    y, V

Proof of Theorem usg2wlkeq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 3anan32 977 . . 3  |-  ( ( N  =  ( # `  ( 1st `  B
) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A ) `
 y )  =  ( ( 1st `  B
) `  y )  /\  A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
) )  <->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
) ) )
21a1i 11 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
)  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  <-> 
( ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
) ) ) )
3 2wlkeq 30412 . . . 4  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
)  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) ) ) )
433expa 1188 . . 3  |-  ( ( ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
)  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) ) ) )
543adant1 1006 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
)  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) ) ) )
6 fzofzp1 11711 . . . . . . . . . . . 12  |-  ( x  e.  ( 0..^ N )  ->  ( x  +  1 )  e.  ( 0 ... N
) )
76adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  x  e.  ( 0..^ N ) )  ->  ( x  +  1 )  e.  ( 0 ... N
) )
8 fveq2 5775 . . . . . . . . . . . . 13  |-  ( y  =  ( x  + 
1 )  ->  (
( 2nd `  A
) `  y )  =  ( ( 2nd `  A ) `  (
x  +  1 ) ) )
9 fveq2 5775 . . . . . . . . . . . . 13  |-  ( y  =  ( x  + 
1 )  ->  (
( 2nd `  B
) `  y )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
108, 9eqeq12d 2471 . . . . . . . . . . . 12  |-  ( y  =  ( x  + 
1 )  ->  (
( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  <->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
1110adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  (
# `  ( 1st `  A ) ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  x  e.  ( 0..^ N ) )  /\  y  =  ( x  +  1 ) )  ->  ( (
( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  <->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
127, 11rspcdv 3158 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  x  e.  ( 0..^ N ) )  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  ->  ( ( 2nd `  A ) `  ( x  +  1
) )  =  ( ( 2nd `  B
) `  ( x  +  1 ) ) ) )
1312impancom 440 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  ->  ( x  e.  ( 0..^ N )  ->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
1413ralrimiv 2880 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  ->  A. x  e.  ( 0..^ N ) ( ( 2nd `  A
) `  ( x  +  1 ) )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
15 oveq1 6183 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
1615fveq2d 5779 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  A ) `  (
x  +  1 ) ) )
1715fveq2d 5779 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( 2nd `  B
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
1816, 17eqeq12d 2471 . . . . . . . . 9  |-  ( y  =  x  ->  (
( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) )  <->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
1918cbvralv 3029 . . . . . . . 8  |-  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  A ) `
 ( y  +  1 ) )  =  ( ( 2nd `  B
) `  ( y  +  1 ) )  <->  A. x  e.  (
0..^ N ) ( ( 2nd `  A
) `  ( x  +  1 ) )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
2014, 19sylibr 212 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  ->  A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) ) )
21 fzossfz 11657 . . . . . . . . . 10  |-  ( 0..^ N )  C_  (
0 ... N )
22 ssralv 3500 . . . . . . . . . 10  |-  ( ( 0..^ N )  C_  ( 0 ... N
)  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  ->  A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
) ) )
2321, 22mp1i 12 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  ->  A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
) ) )
24 r19.26 2931 . . . . . . . . . . 11  |-  ( A. y  e.  ( 0..^ N ) ( ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  /\  ( ( 2nd `  A ) `  ( y  +  1 ) )  =  ( ( 2nd `  B
) `  ( y  +  1 ) ) )  <->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  /\  A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) ) ) )
25 preq12 4040 . . . . . . . . . . . . 13  |-  ( ( ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  /\  ( ( 2nd `  A ) `  ( y  +  1 ) )  =  ( ( 2nd `  B
) `  ( y  +  1 ) ) )  ->  { (
( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } )
2625a1i 11 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( (
( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  /\  ( ( 2nd `  A ) `  ( y  +  1 ) )  =  ( ( 2nd `  B
) `  ( y  +  1 ) ) )  ->  { (
( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } ) )
2726ralimdv 2876 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) ( ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  /\  ( ( 2nd `  A ) `  ( y  +  1 ) )  =  ( ( 2nd `  B
) `  ( y  +  1 ) ) )  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) )
2824, 27syl5bir 218 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( ( A. y  e.  (
0..^ N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  /\  A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) ) )  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) )
2928expd 436 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  A ) `
 y )  =  ( ( 2nd `  B
) `  y )  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) )  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) ) )
3023, 29syld 44 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  A ) `
 ( y  +  1 ) )  =  ( ( 2nd `  B
) `  ( y  +  1 ) )  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } ) ) )
3130imp 429 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) )  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) )
3220, 31mpd 15 . . . . . 6  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } )
3332ex 434 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  ->  A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) )
34 eqid 2450 . . . . . . . . . 10  |-  ( 1st `  A )  =  ( 1st `  A )
35 eqid 2450 . . . . . . . . . 10  |-  ( 2nd `  A )  =  ( 2nd `  A )
3634, 35wlkcompim 30411 . . . . . . . . 9  |-  ( A  e.  ( V Walks  E
)  ->  ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A
) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( E `  ( ( 1st `  A ) `
 y ) )  =  { ( ( 2nd `  A ) `
 y ) ,  ( ( 2nd `  A
) `  ( y  +  1 ) ) } ) )
37 eqid 2450 . . . . . . . . . 10  |-  ( 1st `  B )  =  ( 1st `  B )
38 eqid 2450 . . . . . . . . . 10  |-  ( 2nd `  B )  =  ( 2nd `  B )
3937, 38wlkcompim 30411 . . . . . . . . 9  |-  ( B  e.  ( V Walks  E
)  ->  ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B
) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  B
) ) ) ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) )
40 oveq2 6184 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( 1st `  B ) )  =  N  ->  ( 0..^ ( # `  ( 1st `  B ) ) )  =  ( 0..^ N ) )
4140eqcoms 2461 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( 0..^ (
# `  ( 1st `  B ) ) )  =  ( 0..^ N ) )
4241raleqdv 3005 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( A. y  e.  ( 0..^ ( # `  ( 1st `  B
) ) ) ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  <->  A. y  e.  ( 0..^ N ) ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) } ) )
43 oveq2 6184 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( 1st `  A ) )  =  N  ->  ( 0..^ ( # `  ( 1st `  A ) ) )  =  ( 0..^ N ) )
4443eqcoms 2461 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ (
# `  ( 1st `  A ) ) )  =  ( 0..^ N ) )
4544raleqdv 3005 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( A. y  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( E `  ( ( 1st `  A ) `
 y ) )  =  { ( ( 2nd `  A ) `
 y ) ,  ( ( 2nd `  A
) `  ( y  +  1 ) ) }  <->  A. y  e.  ( 0..^ N ) ( E `  ( ( 1st `  A ) `
 y ) )  =  { ( ( 2nd `  A ) `
 y ) ,  ( ( 2nd `  A
) `  ( y  +  1 ) ) } ) )
4642, 45bi2anan9r 869 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( ( A. y  e.  (
0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  <->  ( A. y  e.  ( 0..^ N ) ( E `
 ( ( 1st `  B ) `  y
) )  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  /\  A. y  e.  ( 0..^ N ) ( E `
 ( ( 1st `  A ) `  y
) )  =  {
( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } ) ) )
47 r19.26 2931 . . . . . . . . . . . . . . . . 17  |-  ( A. y  e.  ( 0..^ N ) ( ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  /\  ( E `
 ( ( 1st `  A ) `  y
) )  =  {
( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  <->  ( A. y  e.  ( 0..^ N ) ( E `
 ( ( 1st `  B ) `  y
) )  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  /\  A. y  e.  ( 0..^ N ) ( E `
 ( ( 1st `  A ) `  y
) )  =  {
( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } ) )
48 eqeq2 2464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  (
( E `  (
( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  <->  ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } ) )
49 eqeq2 2464 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  =  ( E `  ( ( 1st `  A ) `
 y ) )  ->  ( ( E `
 ( ( 1st `  B ) `  y
) )  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  <->  ( E `  ( ( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) )
5049eqcoms 2461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( E `  ( ( 1st `  A ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  ( ( E `  ( ( 1st `  B ) `  y ) )  =  { ( ( 2nd `  B ) `  y
) ,  ( ( 2nd `  B ) `
 ( y  +  1 ) ) }  <-> 
( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) )
5150biimpd 207 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E `  ( ( 1st `  A ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  ( ( E `  ( ( 1st `  B ) `  y ) )  =  { ( ( 2nd `  B ) `  y
) ,  ( ( 2nd `  B ) `
 ( y  +  1 ) ) }  ->  ( E `  ( ( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) )
5248, 51syl6bi 228 . . . . . . . . . . . . . . . . . . . 20  |-  ( { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  (
( E `  (
( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  ->  (
( E `  (
( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  ( E `  ( ( 1st `  B ) `  y ) )  =  ( E `  (
( 1st `  A
) `  y )
) ) ) )
5352com13 80 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  ( ( E `  ( ( 1st `  A ) `  y ) )  =  { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  ->  ( { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  ( E `  ( ( 1st `  B ) `  y ) )  =  ( E `  (
( 1st `  A
) `  y )
) ) ) )
5453imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( E `  (
( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  /\  ( E `  ( ( 1st `  A ) `  y ) )  =  { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) } )  ->  ( {
( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  ( E `  ( ( 1st `  B ) `  y ) )  =  ( E `  (
( 1st `  A
) `  y )
) ) )
5554ral2imi 2802 . . . . . . . . . . . . . . . . 17  |-  ( A. y  e.  ( 0..^ N ) ( ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  /\  ( E `
 ( ( 1st `  A ) `  y
) )  =  {
( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  -> 
( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) )
5647, 55sylbir 213 . . . . . . . . . . . . . . . 16  |-  ( ( A. y  e.  ( 0..^ N ) ( E `  ( ( 1st `  B ) `
 y ) )  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  /\  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  -> 
( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) )
5746, 56syl6bi 228 . . . . . . . . . . . . . . 15  |-  ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( ( A. y  e.  (
0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  -> 
( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) )
5857com12 31 . . . . . . . . . . . . . 14  |-  ( ( A. y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  -> 
( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) )
5958ex 434 . . . . . . . . . . . . 13  |-  ( A. y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  ( A. y  e.  (
0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  ->  (
( N  =  (
# `  ( 1st `  A ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) ) )
60593ad2ant3 1011 . . . . . . . . . . . 12  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } )  -> 
( A. y  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( E `  ( ( 1st `  A ) `
 y ) )  =  { ( ( 2nd `  A ) `
 y ) ,  ( ( 2nd `  A
) `  ( y  +  1 ) ) }  ->  ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) ) )
6160com12 31 . . . . . . . . . . 11  |-  ( A. y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  ->  (
( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } )  -> 
( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) ) )
62613ad2ant3 1011 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  -> 
( ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B
) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } )  -> 
( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) ) )
6362imp 429 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( E `  ( ( 1st `  A
) `  y )
)  =  { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) } )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V  /\  A. y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ( E `  ( ( 1st `  B
) `  y )
)  =  { ( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) } ) )  ->  ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) )
6436, 39, 63syl2an 477 . . . . . . . 8  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) )
6564expd 436 . . . . . . 7  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( N  =  ( # `  ( 1st `  A ) )  ->  ( N  =  ( # `  ( 1st `  B ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) ) )
6665a1i 11 . . . . . 6  |-  ( V USGrph  E  ->  ( ( A  e.  ( V Walks  E
)  /\  B  e.  ( V Walks  E ) )  ->  ( N  =  ( # `  ( 1st `  A ) )  ->  ( N  =  ( # `  ( 1st `  B ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A ) `  y
) ,  ( ( 2nd `  A ) `
 ( y  +  1 ) ) }  =  { ( ( 2nd `  B ) `
 y ) ,  ( ( 2nd `  B
) `  ( y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) ) ) ) )
67663imp1 1201 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) { ( ( 2nd `  A
) `  y ) ,  ( ( 2nd `  A ) `  (
y  +  1 ) ) }  =  {
( ( 2nd `  B
) `  y ) ,  ( ( 2nd `  B ) `  (
y  +  1 ) ) }  ->  A. y  e.  ( 0..^ N ) ( E `  (
( 1st `  B
) `  y )
)  =  ( E `
 ( ( 1st `  A ) `  y
) ) ) )
68 eqcom 2458 . . . . . . 7  |-  ( ( E `  ( ( 1st `  B ) `
 y ) )  =  ( E `  ( ( 1st `  A
) `  y )
)  <->  ( E `  ( ( 1st `  A
) `  y )
)  =  ( E `
 ( ( 1st `  B ) `  y
) ) )
69 usgraf1 23403 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
70693ad2ant1 1009 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  E : dom  E -1-1-> ran  E )
7170adantr 465 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  E : dom  E -1-1-> ran  E )
7271adantr 465 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  E : dom  E -1-1-> ran  E )
73 wlkelwrd 30404 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( V Walks  E
)  ->  ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A
) ) ) --> V ) )
74 wlkelwrd 30404 . . . . . . . . . . . . . . 15  |-  ( B  e.  ( V Walks  E
)  ->  ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B
) ) ) --> V ) )
75 oveq2 6184 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  A ) ) ) )
7675eleq2d 2519 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( y  e.  ( 0..^ N )  <-> 
y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ) )
77 wrdsymbcl 12334 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 1st `  A
)  e. Word  dom  E  /\  y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) )  ->  (
( 1st `  A
) `  y )  e.  dom  E )
7877expcom 435 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( y  e.  ( 0..^ (
# `  ( 1st `  A ) ) )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) )
7976, 78syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) ) )
8079adantr 465 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) ) )
8180imp 429 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  =  (
# `  ( 1st `  A ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y )  e.  dom  E ) )
8281com12 31 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) )
8382adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  ( 1st `  A )  e. Word  dom  E )  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) )
84 oveq2 6184 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  B ) ) ) )
8584eleq2d 2519 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  <-> 
y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ) )
86 wrdsymbcl 12334 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) )  ->  (
( 1st `  B
) `  y )  e.  dom  E )
8786expcom 435 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( y  e.  ( 0..^ (
# `  ( 1st `  B ) ) )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) )
8885, 87syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) ) )
8988adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) ) )
9089imp 429 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  =  (
# `  ( 1st `  A ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y )  e.  dom  E ) )
9190com12 31 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) )
9291adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  ( 1st `  A )  e. Word  dom  E )  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) )
9383, 92jcad 533 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  ( 1st `  A )  e. Word  dom  E )  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) )
9493ex 434 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  A
)  e. Word  dom  E  -> 
( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
9594adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V )  -> 
( ( 1st `  A
)  e. Word  dom  E  -> 
( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
9695com12 31 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V )  -> 
( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
9796adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  -> 
( ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B
) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V )  -> 
( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
9897imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  ->  ( ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( (
( 1st `  A
) `  y )  e.  dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) )
9973, 74, 98syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( (
( N  =  (
# `  ( 1st `  A ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( (
( 1st `  A
) `  y )  e.  dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) )
10099expd 436 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
101100expd 436 . . . . . . . . . . . 12  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( N  =  ( # `  ( 1st `  A ) )  ->  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) ) )
102101imp 429 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
1031023adant1 1006 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) ) )
104103imp 429 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( ( 1st `  A ) `
 y )  e. 
dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) ) )
105104imp 429 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( (
( 1st `  A
) `  y )  e.  dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) )
106 f1veqaeq 6057 . . . . . . . 8  |-  ( ( E : dom  E -1-1-> ran 
E  /\  ( (
( 1st `  A
) `  y )  e.  dom  E  /\  (
( 1st `  B
) `  y )  e.  dom  E ) )  ->  ( ( E `
 ( ( 1st `  A ) `  y
) )  =  ( E `  ( ( 1st `  B ) `
 y ) )  ->  ( ( 1st `  A ) `  y
)  =  ( ( 1st `  B ) `
 y ) ) )
10772, 105, 106syl2anc 661 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( E `  ( ( 1st `  A ) `  y ) )  =  ( E `  (
( 1st `  B
) `  y )
)  ->  ( ( 1st `  A ) `  y )  =  ( ( 1st `  B
) `  y )
) )
10868, 107syl5bi 217 . . . . . 6  |-  ( ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( E `  ( ( 1st `  B ) `  y ) )  =  ( E `  (
( 1st `  A
) `  y )
)  ->  ( ( 1st `  A ) `  y )  =  ( ( 1st `  B
) `  y )
) )
109108ralimdva 2875 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0..^ N ) ( E `
 ( ( 1st `  B ) `  y
) )  =  ( E `  ( ( 1st `  A ) `
 y ) )  ->  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
) ) )
11033, 67, 1093syld 55 . . . 4  |-  ( ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  ->  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
) ) )
111110expimpd 603 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  ->  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
) ) )
112111pm4.71d 634 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  <-> 
( ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) )  /\  A. y  e.  ( 0..^ N ) ( ( 1st `  A
) `  y )  =  ( ( 1st `  B ) `  y
) ) ) )
1132, 5, 1123bitr4d 285 1  |-  ( ( V USGrph  E  /\  ( A  e.  ( V Walks  E )  /\  B  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. y  e.  ( 0 ... N
) ( ( 2nd `  A ) `  y
)  =  ( ( 2nd `  B ) `
 y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   A.wral 2792    C_ wss 3412   {cpr 3963   class class class wbr 4376   dom cdm 4924   ran crn 4925   -->wf 5498   -1-1->wf1 5499   ` cfv 5502  (class class class)co 6176   1stc1st 6661   2ndc2nd 6662   0cc0 9369   1c1 9370    + caddc 9372   ...cfz 11524  ..^cfzo 11635   #chash 12190  Word cword 12309   USGrph cusg 23385   Walks cwalk 23526
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-cda 8424  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-hash 12191  df-word 12317  df-usgra 23387  df-wlk 23536
This theorem is referenced by:  usg2wlkeq2  30465  clwlkf1clwwlk  30647
  Copyright terms: Public domain W3C validator