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Theorem usg2wlkeq 25419
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 994 . . 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 25418 . . . 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 1205 . . 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 1023 . 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 12007 . . . . . . . . . . . 12  |-  ( x  e.  ( 0..^ N )  ->  ( x  +  1 )  e.  ( 0 ... N
) )
76adantl 467 . . . . . . . . . . 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 5877 . . . . . . . . . . . . 13  |-  ( y  =  ( x  + 
1 )  ->  (
( 2nd `  A
) `  y )  =  ( ( 2nd `  A ) `  (
x  +  1 ) ) )
9 fveq2 5877 . . . . . . . . . . . . 13  |-  ( y  =  ( x  + 
1 )  ->  (
( 2nd `  B
) `  y )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
108, 9eqeq12d 2444 . . . . . . . . . . . 12  |-  ( y  =  ( x  + 
1 )  ->  (
( ( 2nd `  A
) `  y )  =  ( ( 2nd `  B ) `  y
)  <->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
1110adantl 467 . . . . . . . . . . 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 3185 . . . . . . . . . 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 441 . . . . . . . . 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 2837 . . . . . . . 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 6308 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
1615fveq2d 5881 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  A ) `  (
x  +  1 ) ) )
1715fveq2d 5881 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( 2nd `  B
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
x  +  1 ) ) )
1816, 17eqeq12d 2444 . . . . . . . . 9  |-  ( y  =  x  ->  (
( ( 2nd `  A
) `  ( y  +  1 ) )  =  ( ( 2nd `  B ) `  (
y  +  1 ) )  <->  ( ( 2nd `  A ) `  (
x  +  1 ) )  =  ( ( 2nd `  B ) `
 ( x  + 
1 ) ) ) )
1918cbvralv 3055 . . . . . . . 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 215 . . . . . . 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 11938 . . . . . . . . . 10  |-  ( 0..^ N )  C_  (
0 ... N )
22 ssralv 3525 . . . . . . . . . 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 13 . . . . . . . . 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 2955 . . . . . . . . . . 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 4078 . . . . . . . . . . . . 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 2835 . . . . . . . . . . 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 221 . . . . . . . . . 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 437 . . . . . . . . 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 45 . . . . . . . 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 430 . . . . . . 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 435 . . . . 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 2422 . . . . . . . . . 10  |-  ( 1st `  A )  =  ( 1st `  A )
35 eqid 2422 . . . . . . . . . 10  |-  ( 2nd `  A )  =  ( 2nd `  A )
3634, 35wlkcompim 25237 . . . . . . . . 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 2422 . . . . . . . . . 10  |-  ( 1st `  B )  =  ( 1st `  B )
38 eqid 2422 . . . . . . . . . 10  |-  ( 2nd `  B )  =  ( 2nd `  B )
3937, 38wlkcompim 25237 . . . . . . . . 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 6309 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( 1st `  B ) )  =  N  ->  ( 0..^ ( # `  ( 1st `  B ) ) )  =  ( 0..^ N ) )
4140eqcoms 2434 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( 0..^ (
# `  ( 1st `  B ) ) )  =  ( 0..^ N ) )
4241raleqdv 3031 . . . . . . . . . . . . . . . . 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 6309 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  ( 1st `  A ) )  =  N  ->  ( 0..^ ( # `  ( 1st `  A ) ) )  =  ( 0..^ N ) )
4443eqcoms 2434 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ (
# `  ( 1st `  A ) ) )  =  ( 0..^ N ) )
4544raleqdv 3031 . . . . . . . . . . . . . . . . 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 882 . . . . . . . . . . . . . . . 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 2955 . . . . . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . . . . . . . . . . . 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 2434 . . . . . . . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . . . . 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 83 . . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . 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 1028 . . . . . . . . . . . 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 32 . . . . . . . . . . 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 1028 . . . . . . . . . 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 430 . . . . . . . . 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 479 . . . . . . . 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 437 . . . . . . 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 1218 . . . . 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 2431 . . . . . . 7  |-  ( ( E `  ( ( 1st `  B ) `
 y ) )  =  ( E `  ( ( 1st `  A
) `  y )
)  <->  ( E `  ( ( 1st `  A
) `  y )
)  =  ( E `
 ( ( 1st `  B ) `  y
) ) )
69 usgraf1 25071 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
70693ad2ant1 1026 . . . . . . . . . 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 466 . . . . . . . . 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 466 . . . . . . . 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 25241 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( V Walks  E
)  ->  ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A
) ) ) --> V ) )
74 wlkelwrd 25241 . . . . . . . . . . . . . . 15  |-  ( B  e.  ( V Walks  E
)  ->  ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B
) ) ) --> V ) )
75 oveq2 6309 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  A ) ) ) )
7675eleq2d 2492 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( y  e.  ( 0..^ N )  <-> 
y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ) )
77 wrdsymbcl 12674 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 1st `  A
)  e. Word  dom  E  /\  y  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) )  ->  (
( 1st `  A
) `  y )  e.  dom  E )
7877expcom 436 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( y  e.  ( 0..^ (
# `  ( 1st `  A ) ) )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) )
7976, 78syl6bi 231 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) ) )
8079adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) ) )
8180imp 430 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  =  (
# `  ( 1st `  A ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( 1st `  A ) `  y )  e.  dom  E ) )
8281com12 32 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  A )  e. Word  dom  E  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  A ) `  y
)  e.  dom  E
) )
8382adantl 467 . . . . . . . . . . . . . . . . . . . . 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 6309 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  B ) ) ) )
8584eleq2d 2492 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  <-> 
y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) ) )
86 wrdsymbcl 12674 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 1st `  B
)  e. Word  dom  E  /\  y  e.  ( 0..^ ( # `  ( 1st `  B ) ) ) )  ->  (
( 1st `  B
) `  y )  e.  dom  E )
8786expcom 436 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( y  e.  ( 0..^ (
# `  ( 1st `  B ) ) )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) )
8885, 87syl6bi 231 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  ( 1st `  B ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) ) )
8988adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  =  ( # `  ( 1st `  A
) )  /\  N  =  ( # `  ( 1st `  B ) ) )  ->  ( y  e.  ( 0..^ N )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) ) )
9089imp 430 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  =  (
# `  ( 1st `  A ) )  /\  N  =  ( # `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( 1st `  B ) `  y )  e.  dom  E ) )
9190com12 32 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  B )  e. Word  dom  E  ->  ( ( ( N  =  ( # `  ( 1st `  A ) )  /\  N  =  (
# `  ( 1st `  B ) ) )  /\  y  e.  ( 0..^ N ) )  ->  ( ( 1st `  B ) `  y
)  e.  dom  E
) )
9291adantr 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 `  B ) `  y
)  e.  dom  E
) )
9383, 92jcad 535 . . . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . 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 437 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 1023 . . . . . . . . . 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 430 . . . . . . . . 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 430 . . . . . . . 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 6172 . . . . . . . 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 665 . . . . . . 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 220 . . . . . 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 2833 . . . . 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 57 . . . 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 606 . . 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 638 . 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 288 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775    C_ wss 3436   {cpr 3998   class class class wbr 4420   dom cdm 4849   ran crn 4850   -->wf 5593   -1-1->wf1 5594   ` cfv 5597  (class class class)co 6301   1stc1st 6801   2ndc2nd 6802   0cc0 9539   1c1 9540    + caddc 9542   ...cfz 11784  ..^cfzo 11915   #chash 12514  Word cword 12646   USGrph cusg 25041   Walks cwalk 25209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12515  df-word 12654  df-usgra 25044  df-wlk 25219
This theorem is referenced by:  usg2wlkeq2  25420  clwlkf1clwwlk  25561
  Copyright terms: Public domain W3C validator