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Theorem usg2wlkeq 24540
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 985 . . 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 24539 . . . 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 1196 . . 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 1014 . 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 11889 . . . . . . . . . . . 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 5872 . . . . . . . . . . . . 13  |-  ( y  =  ( x  + 
1 )  ->  (
( 2nd `  A
) `  y )  =  ( ( 2nd `  A ) `  (
x  +  1 ) ) )
9 fveq2 5872 . . . . . . . . . . . . 13  |-  ( y  =  ( x  + 
1 )  ->  (
( 2nd `  B
) `  y )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
108, 9eqeq12d 2489 . . . . . . . . . . . 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 3222 . . . . . . . . . 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 2879 . . . . . . . 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 6302 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
1615fveq2d 5876 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  A ) `  (
x  +  1 ) ) )
1715fveq2d 5876 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( 2nd `  B
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
1816, 17eqeq12d 2489 . . . . . . . . 9  |-  ( y  =  x  ->  (
( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) )  <->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
1918cbvralv 3093 . . . . . . . 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 11826 . . . . . . . . . 10  |-  ( 0..^ N )  C_  (
0 ... N )
22 ssralv 3569 . . . . . . . . . 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 2994 . . . . . . . . . . 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 4114 . . . . . . . . . . . . 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 2877 . . . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( 1st `  A )  =  ( 1st `  A )
35 eqid 2467 . . . . . . . . . 10  |-  ( 2nd `  A )  =  ( 2nd `  A )
3634, 35wlkcompim 24358 . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( 1st `  B )  =  ( 1st `  B )
38 eqid 2467 . . . . . . . . . 10  |-  ( 2nd `  B )  =  ( 2nd `  B )
3937, 38wlkcompim 24358 . . . . . . . . 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 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( 1st `  B ) )  =  N  ->  ( 0..^ ( # `  ( 1st `  B ) ) )  =  ( 0..^ N ) )
4140eqcoms 2479 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( 0..^ (
# `  ( 1st `  B ) ) )  =  ( 0..^ N ) )
4241raleqdv 3069 . . . . . . . . . . . . . . . . 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 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( 1st `  A ) )  =  N  ->  ( 0..^ ( # `  ( 1st `  A ) ) )  =  ( 0..^ N ) )
4443eqcoms 2479 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ (
# `  ( 1st `  A ) ) )  =  ( 0..^ N ) )
4544raleqdv 3069 . . . . . . . . . . . . . . . . 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 872 . . . . . . . . . . . . . . . 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 2994 . . . . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . 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 1019 . . . . . . . . . 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 1209 . . . . 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 2476 . . . . . . 7  |-  ( ( E `  ( ( 1st `  B ) `
 y ) )  =  ( E `  ( ( 1st `  A
) `  y )
)  <->  ( E `  ( ( 1st `  A
) `  y )
)  =  ( E `
 ( ( 1st `  B ) `  y
) ) )
69 usgraf1 24192 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
70693ad2ant1 1017 . . . . . . . . . 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 24362 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( V Walks  E
)  ->  ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A
) ) ) --> V ) )
74 wlkelwrd 24362 . . . . . . . . . . . . . . 15  |-  ( B  e.  ( V Walks  E
)  ->  ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B
) ) ) --> V ) )
75 oveq2 6303 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  A ) ) ) )
7675eleq2d 2537 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( y  e.  ( 0..^ N )  <-> 
y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ) )
77 wrdsymbcl 12539 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6303 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  B ) ) ) )
8584eleq2d 2537 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  <-> 
y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ) )
86 wrdsymbcl 12539 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1014 . . . . . . . . . 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 6167 . . . . . . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   {cpr 4035   class class class wbr 4453   dom cdm 5005   ran crn 5006   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   0cc0 9504   1c1 9505    + caddc 9507   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12514   USGrph cusg 24162   Walks cwalk 24330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12522  df-usgra 24165  df-wlk 24340
This theorem is referenced by:  usg2wlkeq2  24541  clwlkf1clwwlk  24682
  Copyright terms: Public domain W3C validator