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Theorem wlkelwrd 24732
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 24710 . . 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 3109 . . . 4  |-  v  e. 
_V
3 vex 3109 . . . 4  |-  e  e. 
_V
4 wlks 24721 . . . . 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 6298 . . . . 5  |-  ( v Walks 
e )  e.  _V
64, 5syl6eqelr 2551 . . . 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 670 . . 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 5192 . . . . . . . 8  |-  ( e  =  E  ->  dom  e  =  dom  E )
9 wrdeq 12551 . . . . . . . 8  |-  ( dom  e  =  dom  E  -> Word 
dom  e  = Word  dom  E )
108, 9syl 16 . . . . . . 7  |-  ( e  =  E  -> Word  dom  e  = Word  dom  E )
1110eleq2d 2524 . . . . . 6  |-  ( e  =  E  ->  (
f  e. Word  dom  e  <->  f  e. Word  dom 
E ) )
1211adantl 464 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f  e. Word  dom  e 
<->  f  e. Word  dom  E
) )
13 feq3 5697 . . . . . 6  |-  ( v  =  V  ->  (
p : ( 0 ... ( # `  f
) ) --> v  <->  p :
( 0 ... ( # `
 f ) ) --> V ) )
1413adantr 463 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( p : ( 0 ... ( # `  f ) ) --> v  <-> 
p : ( 0 ... ( # `  f
) ) --> V ) )
15 fveq1 5847 . . . . . . . 8  |-  ( e  =  E  ->  (
e `  ( f `  k ) )  =  ( E `  (
f `  k )
) )
1615adantl 464 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e `  (
f `  k )
)  =  ( E `
 ( f `  k ) ) )
1716eqeq1d 2456 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( e `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  (
f `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
1817ralbidv 2893 . . . . 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 1297 . . . 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 4502 . . 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 6493 . 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 2526 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( f  e. Word  dom  E  <->  ( 1st `  W )  e. Word  dom  E ) )
23 fveq2 5848 . . . . . . . 8  |-  ( f  =  ( 1st `  W
)  ->  ( # `  f
)  =  ( # `  ( 1st `  W
) ) )
2423oveq2d 6286 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( 0 ... ( # `  f
) )  =  ( 0 ... ( # `  ( 1st `  W
) ) ) )
2524feq2d 5700 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( p : ( 0 ... ( # `  f
) ) --> V  <->  p :
( 0 ... ( # `
 ( 1st `  W
) ) ) --> V ) )
2623oveq2d 6286 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( 0..^ ( # `  f
) )  =  ( 0..^ ( # `  ( 1st `  W ) ) ) )
27 fveq1 5847 . . . . . . . . 9  |-  ( f  =  ( 1st `  W
)  ->  ( f `  k )  =  ( ( 1st `  W
) `  k )
)
2827fveq2d 5852 . . . . . . . 8  |-  ( f  =  ( 1st `  W
)  ->  ( E `  ( f `  k
) )  =  ( E `  ( ( 1st `  W ) `
 k ) ) )
2928eqeq1d 2456 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( ( E `  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  <->  ( E `  ( ( 1st `  W
) `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
3026, 29raleqbidv 3065 . . . . . 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 1297 . . . . 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 5695 . . . . . 6  |-  ( p  =  ( 2nd `  W
)  ->  ( p : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  <->  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )
33 fveq1 5847 . . . . . . . . 9  |-  ( p  =  ( 2nd `  W
)  ->  ( p `  k )  =  ( ( 2nd `  W
) `  k )
)
34 fveq1 5847 . . . . . . . . 9  |-  ( p  =  ( 2nd `  W
)  ->  ( p `  ( k  +  1 ) )  =  ( ( 2nd `  W
) `  ( k  +  1 ) ) )
3533, 34preq12d 4103 . . . . . . . 8  |-  ( p  =  ( 2nd `  W
)  ->  { (
p `  k ) ,  ( p `  ( k  +  1 ) ) }  =  { ( ( 2nd `  W ) `  k
) ,  ( ( 2nd `  W ) `
 ( k  +  1 ) ) } )
3635eqeq2d 2468 . . . . . . 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 2893 . . . . . 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 1300 . . . . 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 6834 . . . 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 991 . . . 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 1017 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   {cpr 4018   {copab 4496   dom cdm 4988   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   0cc0 9481   1c1 9482    + caddc 9484   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   Walks cwalk 24700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-wlk 24710
This theorem is referenced by:  vfwlkniswwlkn  24908  2wlkeq  24909  usg2wlkeq  24910
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