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Theorem 2wlkeq 25280
Description: Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.)
Assertion
Ref Expression
2wlkeq  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
Distinct variable groups:    x, A    x, B    x, N
Allowed substitution hints:    E( x)    V( x)

Proof of Theorem 2wlkeq
StepHypRef Expression
1 wlkop 25101 . . . . 5  |-  ( A  e.  ( V Walks  E
)  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2ndb 6845 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
31, 2sylibr 215 . . . 4  |-  ( A  e.  ( V Walks  E
)  ->  A  e.  ( _V  X.  _V )
)
4 wlkop 25101 . . . . 5  |-  ( B  e.  ( V Walks  E
)  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
5 1st2ndb 6845 . . . . 5  |-  ( B  e.  ( _V  X.  _V )  <->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. )
64, 5sylibr 215 . . . 4  |-  ( B  e.  ( V Walks  E
)  ->  B  e.  ( _V  X.  _V )
)
7 xpopth 6846 . . . . 5  |-  ( ( A  e.  ( _V 
X.  _V )  /\  B  e.  ( _V  X.  _V ) )  ->  (
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  A  =  B ) )
87bicomd 204 . . . 4  |-  ( ( A  e.  ( _V 
X.  _V )  /\  B  e.  ( _V  X.  _V ) )  ->  ( A  =  B  <->  ( ( 1st `  A )  =  ( 1st `  B
)  /\  ( 2nd `  A )  =  ( 2nd `  B ) ) ) )
93, 6, 8syl2an 479 . . 3  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( A  =  B  <->  ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A
)  =  ( 2nd `  B ) ) ) )
1093adant3 1025 . 2  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A
)  =  ( 2nd `  B ) ) ) )
11 wlkelwrd 25103 . . . . . 6  |-  ( A  e.  ( V Walks  E
)  ->  ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A
) ) ) --> V ) )
12 wlkelwrd 25103 . . . . . 6  |-  ( B  e.  ( V Walks  E
)  ->  ( ( 1st `  B )  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B
) ) ) --> V ) )
1311, 12anim12i 568 . . . . 5  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( (
( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) ) )
14 eleq1 2501 . . . . . . . 8  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  ( A  e.  ( V Walks  E )  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  ( V Walks  E ) ) )
15 df-br 4427 . . . . . . . . 9  |-  ( ( 1st `  A ) ( V Walks  E ) ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  ( V Walks  E ) )
16 wlklenvm1 25105 . . . . . . . . 9  |-  ( ( 1st `  A ) ( V Walks  E ) ( 2nd `  A
)  ->  ( # `  ( 1st `  A ) )  =  ( ( # `  ( 2nd `  A
) )  -  1 ) )
1715, 16sylbir 216 . . . . . . . 8  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  ( V Walks  E )  -> 
( # `  ( 1st `  A ) )  =  ( ( # `  ( 2nd `  A ) )  -  1 ) )
1814, 17syl6bi 231 . . . . . . 7  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  ( A  e.  ( V Walks  E )  ->  ( # `  ( 1st `  A ) )  =  ( ( # `  ( 2nd `  A
) )  -  1 ) ) )
191, 18mpcom 37 . . . . . 6  |-  ( A  e.  ( V Walks  E
)  ->  ( # `  ( 1st `  A ) )  =  ( ( # `  ( 2nd `  A
) )  -  1 ) )
20 eleq1 2501 . . . . . . . 8  |-  ( B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  ->  ( B  e.  ( V Walks  E )  <->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  ( V Walks  E ) ) )
21 df-br 4427 . . . . . . . . 9  |-  ( ( 1st `  B ) ( V Walks  E ) ( 2nd `  B
)  <->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  ( V Walks  E ) )
22 wlklenvm1 25105 . . . . . . . . 9  |-  ( ( 1st `  B ) ( V Walks  E ) ( 2nd `  B
)  ->  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B
) )  -  1 ) )
2321, 22sylbir 216 . . . . . . . 8  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  ( V Walks  E )  -> 
( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B ) )  -  1 ) )
2420, 23syl6bi 231 . . . . . . 7  |-  ( B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  ->  ( B  e.  ( V Walks  E )  ->  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B
) )  -  1 ) ) )
254, 24mpcom 37 . . . . . 6  |-  ( B  e.  ( V Walks  E
)  ->  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B
) )  -  1 ) )
2619, 25anim12i 568 . . . . 5  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( ( # `
 ( 1st `  A
) )  =  ( ( # `  ( 2nd `  A ) )  -  1 )  /\  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B ) )  -  1 ) ) )
27 eqwrd 12695 . . . . . . . 8  |-  ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 1st `  B )  e. Word  dom  E )  ->  ( ( 1st `  A
)  =  ( 1st `  B )  <->  ( ( # `
 ( 1st `  A
) )  =  (
# `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( ( 1st `  A ) `  x
)  =  ( ( 1st `  B ) `
 x ) ) ) )
2827ad2ant2r 751 . . . . . . 7  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  ->  ( ( 1st `  A )  =  ( 1st `  B )  <-> 
( ( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) ) ) )
2928adantr 466 . . . . . 6  |-  ( ( ( ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  /\  ( ( # `  ( 1st `  A
) )  =  ( ( # `  ( 2nd `  A ) )  -  1 )  /\  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B ) )  -  1 ) ) )  ->  ( ( 1st `  A )  =  ( 1st `  B
)  <->  ( ( # `  ( 1st `  A
) )  =  (
# `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( ( 1st `  A ) `  x
)  =  ( ( 1st `  B ) `
 x ) ) ) )
30 lencl 12674 . . . . . . . . . . 11  |-  ( ( 1st `  A )  e. Word  dom  E  ->  (
# `  ( 1st `  A ) )  e. 
NN0 )
3130adantr 466 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  -> 
( # `  ( 1st `  A ) )  e. 
NN0 )
3231adantr 466 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  ->  ( # `  ( 1st `  A ) )  e.  NN0 )
33 simplr 760 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  ->  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )
34 simprr 764 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  ->  ( 2nd `  B
) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V )
3532, 33, 343jca 1185 . . . . . . . 8  |-  ( ( ( ( 1st `  A
)  e. Word  dom  E  /\  ( 2nd `  A ) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  ->  ( ( # `  ( 1st `  A
) )  e.  NN0  /\  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V  /\  ( 2nd `  B ) : ( 0 ... ( # `
 ( 1st `  B
) ) ) --> V ) )
3635adantr 466 . . . . . . 7  |-  ( ( ( ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  /\  ( ( # `  ( 1st `  A
) )  =  ( ( # `  ( 2nd `  A ) )  -  1 )  /\  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B ) )  -  1 ) ) )  ->  ( ( # `
 ( 1st `  A
) )  e.  NN0  /\  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V  /\  ( 2nd `  B ) : ( 0 ... ( # `
 ( 1st `  B
) ) ) --> V ) )
37 2ffzeq 11908 . . . . . . 7  |-  ( ( ( # `  ( 1st `  A ) )  e.  NN0  /\  ( 2nd `  A ) : ( 0 ... ( # `
 ( 1st `  A
) ) ) --> V  /\  ( 2nd `  B
) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V )  -> 
( ( 2nd `  A
)  =  ( 2nd `  B )  <->  ( ( # `
 ( 1st `  A
) )  =  (
# `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `  ( 1st `  A ) ) ) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
3836, 37syl 17 . . . . . 6  |-  ( ( ( ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  /\  ( ( # `  ( 1st `  A
) )  =  ( ( # `  ( 2nd `  A ) )  -  1 )  /\  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B ) )  -  1 ) ) )  ->  ( ( 2nd `  A )  =  ( 2nd `  B
)  <->  ( ( # `  ( 1st `  A
) )  =  (
# `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `  ( 1st `  A ) ) ) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
3929, 38anbi12d 715 . . . . 5  |-  ( ( ( ( ( 1st `  A )  e. Word  dom  E  /\  ( 2nd `  A
) : ( 0 ... ( # `  ( 1st `  A ) ) ) --> V )  /\  ( ( 1st `  B
)  e. Word  dom  E  /\  ( 2nd `  B ) : ( 0 ... ( # `  ( 1st `  B ) ) ) --> V ) )  /\  ( ( # `  ( 1st `  A
) )  =  ( ( # `  ( 2nd `  A ) )  -  1 )  /\  ( # `  ( 1st `  B ) )  =  ( ( # `  ( 2nd `  B ) )  -  1 ) ) )  ->  ( (
( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  ( (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) ) )
4013, 26, 39syl2anc 665 . . . 4  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E ) )  ->  ( (
( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  ( (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) ) )
41403adant3 1025 . . 3  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( ( ( 1st `  A )  =  ( 1st `  B
)  /\  ( 2nd `  A )  =  ( 2nd `  B ) )  <->  ( ( (
# `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) ) )
42 eqeq1 2433 . . . . . . 7  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( N  =  ( # `  ( 1st `  B ) )  <-> 
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) ) ) )
43 oveq2 6313 . . . . . . . 8  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0..^ N )  =  ( 0..^ ( # `  ( 1st `  A ) ) ) )
4443raleqdv 3038 . . . . . . 7  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  <->  A. x  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( ( 1st `  A ) `  x
)  =  ( ( 1st `  B ) `
 x ) ) )
4542, 44anbi12d 715 . . . . . 6  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  <->  ( ( # `
 ( 1st `  A
) )  =  (
# `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A ) ) ) ( ( 1st `  A ) `  x
)  =  ( ( 1st `  B ) `
 x ) ) ) )
46 oveq2 6313 . . . . . . . 8  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( 0 ... N )  =  ( 0 ... ( # `  ( 1st `  A
) ) ) )
4746raleqdv 3038 . . . . . . 7  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x )  <->  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) )
4842, 47anbi12d 715 . . . . . 6  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) )  <-> 
( ( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) )
4945, 48anbi12d 715 . . . . 5  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( ( ( N  =  ( # `  ( 1st `  B
) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A ) `
 x )  =  ( ( 1st `  B
) `  x )
)  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) )  <->  ( ( (
# `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) ) )
5049bibi2d 319 . . . 4  |-  ( N  =  ( # `  ( 1st `  A ) )  ->  ( ( ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )  <->  ( (
( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  ( (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) ) ) )
51503ad2ant3 1028 . . 3  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( ( ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )  <->  ( (
( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) )  <->  ( (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ ( # `  ( 1st `  A
) ) ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  (
( # `  ( 1st `  A ) )  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... ( # `
 ( 1st `  A
) ) ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) ) ) ) ) )
5241, 51mpbird 235 . 2  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( ( ( 1st `  A )  =  ( 1st `  B
)  /\  ( 2nd `  A )  =  ( 2nd `  B ) )  <->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) ) )
53 3anass 986 . . . 4  |-  ( ( N  =  ( # `  ( 1st `  B
) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A ) `
 x )  =  ( ( 1st `  B
) `  x )  /\  A. x  e.  ( 0 ... N ) ( ( 2nd `  A
) `  x )  =  ( ( 2nd `  B ) `  x
) )  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  ( A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
54 anandi 835 . . . 4  |-  ( ( N  =  ( # `  ( 1st `  B
) )  /\  ( A. x  e.  (
0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) )  <->  ( ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
) )  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
5553, 54bitr2i 253 . . 3  |-  ( ( ( N  =  (
# `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A ) `
 x )  =  ( ( 1st `  B
) `  x )
)  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) )  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) )
5655a1i 11 . 2  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( ( ( N  =  ( # `  ( 1st `  B
) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A ) `
 x )  =  ( ( 1st `  B
) `  x )
)  /\  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) )  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
5710, 52, 563bitrd 282 1  |-  ( ( A  e.  ( V Walks 
E )  /\  B  e.  ( V Walks  E )  /\  N  =  (
# `  ( 1st `  A ) ) )  ->  ( A  =  B  <->  ( N  =  ( # `  ( 1st `  B ) )  /\  A. x  e.  ( 0..^ N ) ( ( 1st `  A
) `  x )  =  ( ( 1st `  B ) `  x
)  /\  A. x  e.  ( 0 ... N
) ( ( 2nd `  A ) `  x
)  =  ( ( 2nd `  B ) `
 x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087   <.cop 4008   class class class wbr 4426    X. cxp 4852   dom cdm 4854   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   0cc0 9538   1c1 9539    - cmin 9859   NN0cn0 10869   ...cfz 11782  ..^cfzo 11913   #chash 12512  Word cword 12643   Walks cwalk 25071
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-wlk 25081
This theorem is referenced by:  usg2wlkeq  25281
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