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Theorem wlkelwrd 30233
Description: The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Assertion
Ref Expression
wlkelwrd  |-  ( W  e.  ( V Walks  E
)  ->  ( ( 1st `  W )  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )

Proof of Theorem wlkelwrd
Dummy variables  e 
f  k  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlk 23366 . . 3  |- Walks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f ) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( e `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
2 vex 2970 . . . 4  |-  v  e. 
_V
3 vex 2970 . . . 4  |-  e  e. 
_V
4 wlks 23376 . . . . 5  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v Walks  e )  =  { <. f ,  p >.  |  (
f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) } )
5 ovex 6111 . . . . 5  |-  ( v Walks 
e )  e.  _V
64, 5syl6eqelr 2527 . . . 4  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  e.  _V )
72, 3, 6mp2an 672 . . 3  |-  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  e.  _V
8 dmeq 5035 . . . . . . . 8  |-  ( e  =  E  ->  dom  e  =  dom  E )
9 wrdeq 12243 . . . . . . . 8  |-  ( dom  e  =  dom  E  -> Word 
dom  e  = Word  dom  E )
108, 9syl 16 . . . . . . 7  |-  ( e  =  E  -> Word  dom  e  = Word  dom  E )
1110eleq2d 2505 . . . . . 6  |-  ( e  =  E  ->  (
f  e. Word  dom  e  <->  f  e. Word  dom 
E ) )
1211adantl 466 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f  e. Word  dom  e 
<->  f  e. Word  dom  E
) )
13 feq3 5539 . . . . . 6  |-  ( v  =  V  ->  (
p : ( 0 ... ( # `  f
) ) --> v  <->  p :
( 0 ... ( # `
 f ) ) --> V ) )
1413adantr 465 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( p : ( 0 ... ( # `  f ) ) --> v  <-> 
p : ( 0 ... ( # `  f
) ) --> V ) )
15 fveq1 5685 . . . . . . . 8  |-  ( e  =  E  ->  (
e `  ( f `  k ) )  =  ( E `  (
f `  k )
) )
1615adantl 466 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e `  (
f `  k )
)  =  ( E `
 ( f `  k ) ) )
1716eqeq1d 2446 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( e `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  (
f `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
1817ralbidv 2730 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( A. k  e.  ( 0..^ ( # `  f ) ) ( e `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) }  <->  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )
1912, 14, 183anbi123d 1289 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  <->  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
2019opabbidv 4350 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  =  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
211, 7, 20elovmpt2 6302 . 2  |-  ( W  e.  ( V Walks  E
)  <->  ( V  e. 
_V  /\  E  e.  _V  /\  W  e.  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } ) )
22 eleq1 2498 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( f  e. Word  dom  E  <->  ( 1st `  W )  e. Word  dom  E ) )
23 fveq2 5686 . . . . . . . 8  |-  ( f  =  ( 1st `  W
)  ->  ( # `  f
)  =  ( # `  ( 1st `  W
) ) )
2423oveq2d 6102 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( 0 ... ( # `  f
) )  =  ( 0 ... ( # `  ( 1st `  W
) ) ) )
2524feq2d 5542 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( p : ( 0 ... ( # `  f
) ) --> V  <->  p :
( 0 ... ( # `
 ( 1st `  W
) ) ) --> V ) )
2623oveq2d 6102 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( 0..^ ( # `  f
) )  =  ( 0..^ ( # `  ( 1st `  W ) ) ) )
27 fveq1 5685 . . . . . . . . 9  |-  ( f  =  ( 1st `  W
)  ->  ( f `  k )  =  ( ( 1st `  W
) `  k )
)
2827fveq2d 5690 . . . . . . . 8  |-  ( f  =  ( 1st `  W
)  ->  ( E `  ( f `  k
) )  =  ( E `  ( ( 1st `  W ) `
 k ) ) )
2928eqeq1d 2446 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( ( E `  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  <->  ( E `  ( ( 1st `  W
) `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
3026, 29raleqbidv 2926 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  <->  A. k  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
3122, 25, 303anbi123d 1289 . . . . 5  |-  ( f  =  ( 1st `  W
)  ->  ( (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  <->  ( ( 1st `  W )  e. Word  dom  E  /\  p : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  ( 1st `  W
) ) ) ( E `  ( ( 1st `  W ) `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) )
32 feq1 5537 . . . . . 6  |-  ( p  =  ( 2nd `  W
)  ->  ( p : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  <->  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )
33 fveq1 5685 . . . . . . . . 9  |-  ( p  =  ( 2nd `  W
)  ->  ( p `  k )  =  ( ( 2nd `  W
) `  k )
)
34 fveq1 5685 . . . . . . . . 9  |-  ( p  =  ( 2nd `  W
)  ->  ( p `  ( k  +  1 ) )  =  ( ( 2nd `  W
) `  ( k  +  1 ) ) )
3533, 34preq12d 3957 . . . . . . . 8  |-  ( p  =  ( 2nd `  W
)  ->  { (
p `  k ) ,  ( p `  ( k  +  1 ) ) }  =  { ( ( 2nd `  W ) `  k
) ,  ( ( 2nd `  W ) `
 ( k  +  1 ) ) } )
3635eqeq2d 2449 . . . . . . 7  |-  ( p  =  ( 2nd `  W
)  ->  ( ( E `  ( ( 1st `  W ) `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  <->  ( E `  ( ( 1st `  W
) `  k )
)  =  { ( ( 2nd `  W
) `  k ) ,  ( ( 2nd `  W ) `  (
k  +  1 ) ) } ) )
3736ralbidv 2730 . . . . . 6  |-  ( p  =  ( 2nd `  W
)  ->  ( A. k  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  <->  A. k  e.  ( 0..^ ( # `  ( 1st `  W
) ) ) ( E `  ( ( 1st `  W ) `
 k ) )  =  { ( ( 2nd `  W ) `
 k ) ,  ( ( 2nd `  W
) `  ( k  +  1 ) ) } ) )
3832, 373anbi23d 1292 . . . . 5  |-  ( p  =  ( 2nd `  W
)  ->  ( (
( 1st `  W
)  e. Word  dom  E  /\  p : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  <-> 
( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  k )
)  =  { ( ( 2nd `  W
) `  k ) ,  ( ( 2nd `  W ) `  (
k  +  1 ) ) } ) ) )
3931, 38elopabi 6630 . . . 4  |-  ( W  e.  { <. f ,  p >.  |  (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  ->  ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  k )
)  =  { ( ( 2nd `  W
) `  k ) ,  ( ( 2nd `  W ) `  (
k  +  1 ) ) } ) )
40 3simpa 985 . . . 4  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  k )
)  =  { ( ( 2nd `  W
) `  k ) ,  ( ( 2nd `  W ) `  (
k  +  1 ) ) } )  -> 
( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V ) )
4139, 40syl 16 . . 3  |-  ( W  e.  { <. f ,  p >.  |  (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  ->  ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V ) )
42413ad2ant3 1011 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )  ->  ( ( 1st `  W )  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )
4321, 42sylbi 195 1  |-  ( W  e.  ( V Walks  E
)  ->  ( ( 1st `  W )  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967   {cpr 3874   {copab 4344   dom cdm 4835   -->wf 5409   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   0cc0 9274   1c1 9275    + caddc 9277   ...cfz 11429  ..^cfzo 11540   #chash 12095  Word cword 12213   Walks cwalk 23356
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-word 12221  df-wlk 23366
This theorem is referenced by:  2wlkeq  30241  usg2wlkeq  30242  vfwlkniswwlkn  30293
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