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Theorem wlkelwrd 30421
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 23560 . . 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 3074 . . . 4  |-  v  e. 
_V
3 vex 3074 . . . 4  |-  e  e. 
_V
4 wlks 23570 . . . . 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 6218 . . . . 5  |-  ( v Walks 
e )  e.  _V
64, 5syl6eqelr 2548 . . . 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 5141 . . . . . . . 8  |-  ( e  =  E  ->  dom  e  =  dom  E )
9 wrdeq 12362 . . . . . . . 8  |-  ( dom  e  =  dom  E  -> Word 
dom  e  = Word  dom  E )
108, 9syl 16 . . . . . . 7  |-  ( e  =  E  -> Word  dom  e  = Word  dom  E )
1110eleq2d 2521 . . . . . 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 5645 . . . . . 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 5791 . . . . . . . 8  |-  ( e  =  E  ->  (
e `  ( f `  k ) )  =  ( E `  (
f `  k )
) )
1615adantl 466 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e `  (
f `  k )
)  =  ( E `
 ( f `  k ) ) )
1716eqeq1d 2453 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( e `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  (
f `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
1817ralbidv 2841 . . . . 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 1290 . . . 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 4456 . . 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 6410 . 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 2523 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( f  e. Word  dom  E  <->  ( 1st `  W )  e. Word  dom  E ) )
23 fveq2 5792 . . . . . . . 8  |-  ( f  =  ( 1st `  W
)  ->  ( # `  f
)  =  ( # `  ( 1st `  W
) ) )
2423oveq2d 6209 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( 0 ... ( # `  f
) )  =  ( 0 ... ( # `  ( 1st `  W
) ) ) )
2524feq2d 5648 . . . . . 6  |-  ( f  =  ( 1st `  W
)  ->  ( p : ( 0 ... ( # `  f
) ) --> V  <->  p :
( 0 ... ( # `
 ( 1st `  W
) ) ) --> V ) )
2623oveq2d 6209 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( 0..^ ( # `  f
) )  =  ( 0..^ ( # `  ( 1st `  W ) ) ) )
27 fveq1 5791 . . . . . . . . 9  |-  ( f  =  ( 1st `  W
)  ->  ( f `  k )  =  ( ( 1st `  W
) `  k )
)
2827fveq2d 5796 . . . . . . . 8  |-  ( f  =  ( 1st `  W
)  ->  ( E `  ( f `  k
) )  =  ( E `  ( ( 1st `  W ) `
 k ) ) )
2928eqeq1d 2453 . . . . . . 7  |-  ( f  =  ( 1st `  W
)  ->  ( ( E `  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  <->  ( E `  ( ( 1st `  W
) `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
3026, 29raleqbidv 3030 . . . . . 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 1290 . . . . 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 5643 . . . . . 6  |-  ( p  =  ( 2nd `  W
)  ->  ( p : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  <->  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )
33 fveq1 5791 . . . . . . . . 9  |-  ( p  =  ( 2nd `  W
)  ->  ( p `  k )  =  ( ( 2nd `  W
) `  k )
)
34 fveq1 5791 . . . . . . . . 9  |-  ( p  =  ( 2nd `  W
)  ->  ( p `  ( k  +  1 ) )  =  ( ( 2nd `  W
) `  ( k  +  1 ) ) )
3533, 34preq12d 4063 . . . . . . . 8  |-  ( p  =  ( 2nd `  W
)  ->  { (
p `  k ) ,  ( p `  ( k  +  1 ) ) }  =  { ( ( 2nd `  W ) `  k
) ,  ( ( 2nd `  W ) `
 ( k  +  1 ) ) } )
3635eqeq2d 2465 . . . . . . 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 2841 . . . . . 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 1293 . . . . 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 6738 . . . 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 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071   {cpr 3980   {copab 4450   dom cdm 4941   -->wf 5515   ` cfv 5519  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679   0cc0 9386   1c1 9387    + caddc 9389   ...cfz 11547  ..^cfzo 11658   #chash 12213  Word cword 12332   Walks cwalk 23550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-word 12340  df-wlk 23560
This theorem is referenced by:  2wlkeq  30429  usg2wlkeq  30430  vfwlkniswwlkn  30481
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