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Theorem wwlkextsur 30389
Description: Lemma 3 for wwlkextbij 30391. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
Ref Expression
wwlkextbij.d  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
wwlkextbij.r  |-  R  =  { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }
wwlkextbij.f  |-  F  =  ( t  e.  D  |->  ( lastS  `  t )
)
Assertion
Ref Expression
wwlkextsur  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  F : D -onto-> R )
Distinct variable groups:    t, D    n, E, w    t, N, w    t, R    n, V, t, w    n, W, t, w
Allowed substitution hints:    D( w, n)    R( w, n)    E( t)    F( w, t, n)    N( n)

Proof of Theorem wwlkextsur
Dummy variables  i 
d  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlknprop 30346 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
2 simprl 755 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) )  ->  N  e.  NN0 )
3 wwlkextbij.d . . . 4  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
4 wwlkextbij.r . . . 4  |-  R  =  { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }
5 wwlkextbij.f . . . 4  |-  F  =  ( t  e.  D  |->  ( lastS  `  t )
)
63, 4, 5wwlkextfun 30387 . . 3  |-  ( N  e.  NN0  ->  F : D
--> R )
71, 2, 63syl 20 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  F : D
--> R )
84eleq2i 2507 . . . . 5  |-  ( r  e.  R  <->  r  e.  { n  e.  V  |  { ( lastS  `  W ) ,  n }  e.  ran  E } )
9 preq2 3976 . . . . . . 7  |-  ( n  =  r  ->  { ( lastS  `  W ) ,  n }  =  { ( lastS  `  W ) ,  r } )
109eleq1d 2509 . . . . . 6  |-  ( n  =  r  ->  ( { ( lastS  `  W ) ,  n }  e.  ran  E  <->  { ( lastS  `  W
) ,  r }  e.  ran  E ) )
1110elrab 3138 . . . . 5  |-  ( r  e.  { n  e.  V  |  { ( lastS  `  W ) ,  n }  e.  ran  E }  <->  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E ) )
128, 11bitri 249 . . . 4  |-  ( r  e.  R  <->  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )
13 wwlknext 30382 . . . . . . . . . . 11  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  r  e.  V  /\  { ( lastS  `  W
) ,  r }  e.  ran  E )  ->  ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )
14133expb 1188 . . . . . . . . . 10  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )
15 wwlknimp 30347 . . . . . . . . . . . . . 14  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( N  +  1 )  /\  A. i  e.  ( 0..^ N ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E
) )
16 s1cl 12314 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e.  V  ->  <" r ">  e. Word  V )
17 swrdccat1 12372 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e. Word  V  /\  <" r ">  e. Word  V )  ->  (
( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. )  =  W )
1816, 17sylan2 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e. Word  V  /\  r  e.  V )  ->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. )  =  W )
1918ex 434 . . . . . . . . . . . . . . . . 17  |-  ( W  e. Word  V  ->  (
r  e.  V  -> 
( ( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. )  =  W ) )
2019adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( r  e.  V  ->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. )  =  W ) )
21 opeq2 4081 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  +  1 )  =  ( # `  W
)  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( # `  W
) >. )
2221eqcoms 2446 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  <. 0 ,  ( N  + 
1 ) >.  =  <. 0 ,  ( # `  W
) >. )
2322oveq2d 6128 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  ( ( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. ) )
2423eqeq1d 2451 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( ( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. )  =  W ) )
2524adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 )
>. )  =  W  <->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( # `  W
) >. )  =  W ) )
2620, 25sylibrd 234 . . . . . . . . . . . . . . 15  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( r  e.  V  ->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
27263adant3 1008 . . . . . . . . . . . . . 14  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 )  /\  A. i  e.  ( 0..^ N ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  (
r  e.  V  -> 
( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
2815, 27syl 16 . . . . . . . . . . . . 13  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( r  e.  V  ->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
2928com12 31 . . . . . . . . . . . 12  |-  ( r  e.  V  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
3029adantr 465 . . . . . . . . . . 11  |-  ( ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E )  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
3130impcom 430 . . . . . . . . . 10  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  (
( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W )
32 lswccats1 12333 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( W  e. Word  V  /\  r  e.  V )  ->  ( lastS  `  ( W concat  <" r "> ) )  =  r )
3332eqcomd 2448 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e. Word  V  /\  r  e.  V )  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) )
3433ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( W  e. Word  V  ->  (
r  e.  V  -> 
r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
3534adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( r  e.  V  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
3635adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) )  ->  ( r  e.  V  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
371, 36syl 16 . . . . . . . . . . . . . . 15  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( r  e.  V  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
3837imp 429 . . . . . . . . . . . . . 14  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  r  e.  V )  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) )
3938preq2d 3982 . . . . . . . . . . . . 13  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  r  e.  V )  ->  { ( lastS  `  W ) ,  r }  =  { ( lastS  `  W ) ,  ( lastS  `  ( W concat  <" r "> ) ) } )
4039eleq1d 2509 . . . . . . . . . . . 12  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  r  e.  V )  ->  ( { ( lastS  `  W ) ,  r }  e.  ran  E  <->  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )
4140biimpd 207 . . . . . . . . . . 11  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  r  e.  V )  ->  ( { ( lastS  `  W ) ,  r }  e.  ran  E  ->  { ( lastS  `  W ) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )
4241impr 619 . . . . . . . . . 10  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  { ( lastS  `  W ) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E )
4314, 31, 42jca32 535 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  (
( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  (
( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) ) )
4437com12 31 . . . . . . . . . . 11  |-  ( r  e.  V  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
4544adantr 465 . . . . . . . . . 10  |-  ( ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E )  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
4645impcom 430 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  r  =  ( lastS  `  ( W concat  <" r "> ) ) )
47 ovex 6137 . . . . . . . . . . 11  |-  ( W concat  <" r "> )  e.  _V
4847a1i 11 . . . . . . . . . 10  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  ( W concat  <" r "> )  e.  _V )
49 eleq1 2503 . . . . . . . . . . . . . . 15  |-  ( d  =  ( W concat  <" r "> )  ->  (
d  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  <->  ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
50 oveq1 6119 . . . . . . . . . . . . . . . . 17  |-  ( d  =  ( W concat  <" r "> )  ->  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  (
( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. ) )
5150eqeq1d 2451 . . . . . . . . . . . . . . . 16  |-  ( d  =  ( W concat  <" r "> )  ->  (
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
52 fveq2 5712 . . . . . . . . . . . . . . . . . 18  |-  ( d  =  ( W concat  <" r "> )  ->  ( lastS  `  d )  =  ( lastS  `  ( W concat  <" r "> ) ) )
5352preq2d 3982 . . . . . . . . . . . . . . . . 17  |-  ( d  =  ( W concat  <" r "> )  ->  { ( lastS  `  W ) ,  ( lastS  `  d ) }  =  { ( lastS  `  W ) ,  ( lastS  `  ( W concat  <" r "> ) ) } )
5453eleq1d 2509 . . . . . . . . . . . . . . . 16  |-  ( d  =  ( W concat  <" r "> )  ->  ( { ( lastS  `  W ) ,  ( lastS  `  d
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )
5551, 54anbi12d 710 . . . . . . . . . . . . . . 15  |-  ( d  =  ( W concat  <" r "> )  ->  (
( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
)  <->  ( ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) ) )
5649, 55anbi12d 710 . . . . . . . . . . . . . 14  |-  ( d  =  ( W concat  <" r "> )  ->  (
( d  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  (
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  <->  ( ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  (
( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) ) ) )
5752eqeq2d 2454 . . . . . . . . . . . . . 14  |-  ( d  =  ( W concat  <" r "> )  ->  (
r  =  ( lastS  `  d
)  <->  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) )
5856, 57anbi12d 710 . . . . . . . . . . . . 13  |-  ( d  =  ( W concat  <" r "> )  ->  (
( ( d  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  r  =  ( lastS  `  d ) )  <->  ( ( ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  (
( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )  /\  r  =  ( lastS  `  ( W concat  <" r "> ) ) ) ) )
5958bicomd 201 . . . . . . . . . . . 12  |-  ( d  =  ( W concat  <" r "> )  ->  (
( ( ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )  /\  r  =  ( lastS  `  ( W concat  <" r "> ) ) )  <->  ( (
d  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  r  =  ( lastS  `  d ) ) ) )
6059adantl 466 . . . . . . . . . . 11  |-  ( ( ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E ) )  /\  d  =  ( W concat  <" r "> ) )  ->  (
( ( ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )  /\  r  =  ( lastS  `  ( W concat  <" r "> ) ) )  <->  ( (
d  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  r  =  ( lastS  `  d ) ) ) )
6160biimpd 207 . . . . . . . . . 10  |-  ( ( ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E ) )  /\  d  =  ( W concat  <" r "> ) )  ->  (
( ( ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )  /\  r  =  ( lastS  `  ( W concat  <" r "> ) ) )  -> 
( ( d  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  r  =  ( lastS  `  d ) ) ) )
6248, 61spcimedv 3077 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  (
( ( ( W concat  <" r "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( ( W concat  <" r "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  ( W concat  <" r "> ) ) }  e.  ran  E ) )  /\  r  =  ( lastS  `  ( W concat  <" r "> ) ) )  ->  E. d ( ( d  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  r  =  ( lastS  `  d ) ) ) )
6343, 46, 62mp2and 679 . . . . . . . 8  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  E. d
( ( d  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  r  =  ( lastS  `  d ) ) )
64 oveq1 6119 . . . . . . . . . . . . 13  |-  ( w  =  d  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
d substr  <. 0 ,  ( N  +  1 )
>. ) )
6564eqeq1d 2451 . . . . . . . . . . . 12  |-  ( w  =  d  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
66 fveq2 5712 . . . . . . . . . . . . . 14  |-  ( w  =  d  ->  ( lastS  `  w )  =  ( lastS  `  d ) )
6766preq2d 3982 . . . . . . . . . . . . 13  |-  ( w  =  d  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  d
) } )
6867eleq1d 2509 . . . . . . . . . . . 12  |-  ( w  =  d  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  d
) }  e.  ran  E ) )
6965, 68anbi12d 710 . . . . . . . . . . 11  |-  ( w  =  d  ->  (
( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  <->  ( (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) ) )
7069elrab 3138 . . . . . . . . . 10  |-  ( d  e.  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  <->  ( d  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) ) )
7170anbi1i 695 . . . . . . . . 9  |-  ( ( d  e.  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  /\  r  =  ( lastS  `  d ) )  <->  ( ( d  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  r  =  ( lastS  `  d ) ) )
7271exbii 1634 . . . . . . . 8  |-  ( E. d ( d  e. 
{ w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }  /\  r  =  ( lastS  `  d
) )  <->  E. d
( ( d  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  r  =  ( lastS  `  d ) ) )
7363, 72sylibr 212 . . . . . . 7  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  E. d
( d  e.  {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  /\  r  =  ( lastS  `  d ) ) )
74 df-rex 2742 . . . . . . 7  |-  ( E. d  e.  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) } r  =  ( lastS  `  d )  <->  E. d ( d  e. 
{ w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }  /\  r  =  ( lastS  `  d
) ) )
7573, 74sylibr 212 . . . . . 6  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  E. d  e.  { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } r  =  ( lastS  `  d
) )
763wwlkextwrd 30386 . . . . . . . 8  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  D  =  { w  e.  (
( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } )
7776adantr 465 . . . . . . 7  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  D  =  { w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } )
7877rexeqdv 2945 . . . . . 6  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  ( E. d  e.  D  r  =  ( lastS  `  d
)  <->  E. d  e.  {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } r  =  ( lastS  `  d )
) )
7975, 78mpbird 232 . . . . 5  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  E. d  e.  D  r  =  ( lastS  `  d ) )
80 fvex 5722 . . . . . . . 8  |-  ( lastS  `  d
)  e.  _V
81 fveq2 5712 . . . . . . . . 9  |-  ( t  =  d  ->  ( lastS  `  t )  =  ( lastS  `  d ) )
8281, 5fvmptg 5793 . . . . . . . 8  |-  ( ( d  e.  D  /\  ( lastS  `  d )  e. 
_V )  ->  ( F `  d )  =  ( lastS  `  d ) )
8380, 82mpan2 671 . . . . . . 7  |-  ( d  e.  D  ->  ( F `  d )  =  ( lastS  `  d ) )
8483eqeq2d 2454 . . . . . 6  |-  ( d  e.  D  ->  (
r  =  ( F `
 d )  <->  r  =  ( lastS  `  d ) ) )
8584rexbiia 2769 . . . . 5  |-  ( E. d  e.  D  r  =  ( F `  d )  <->  E. d  e.  D  r  =  ( lastS  `  d ) )
8679, 85sylibr 212 . . . 4  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( r  e.  V  /\  { ( lastS  `  W ) ,  r }  e.  ran  E
) )  ->  E. d  e.  D  r  =  ( F `  d ) )
8712, 86sylan2b 475 . . 3  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  r  e.  R )  ->  E. d  e.  D  r  =  ( F `  d ) )
8887ralrimiva 2820 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  A. r  e.  R  E. d  e.  D  r  =  ( F `  d ) )
89 dffo3 5879 . 2  |-  ( F : D -onto-> R  <->  ( F : D --> R  /\  A. r  e.  R  E. d  e.  D  r  =  ( F `  d ) ) )
907, 88, 89sylanbrc 664 1  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  F : D -onto-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 2993   {cpr 3900   <.cop 3904    e. cmpt 4371   ran crn 4862   -->wf 5435   -onto->wfo 5437   ` cfv 5439  (class class class)co 6112   0cc0 9303   1c1 9304    + caddc 9306   2c2 10392   NN0cn0 10600  ..^cfzo 11569   #chash 12124  Word cword 12242   lastS clsw 12243   concat cconcat 12244   <"cs1 12245   substr csubstr 12246   WWalksN cwwlkn 30338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-lsw 12251  df-concat 12252  df-s1 12253  df-substr 12254  df-wwlk 30339  df-wwlkn 30340
This theorem is referenced by:  wwlkextbij0  30390
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