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Theorem wlkcompim 30436
Description: Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Hypotheses
Ref Expression
wlkcomp.1  |-  F  =  ( 1st `  W
)
wlkcomp.2  |-  P  =  ( 2nd `  W
)
Assertion
Ref Expression
wlkcompim  |-  ( W  e.  ( V Walks  E
)  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
Distinct variable groups:    k, E    k, F    P, k    k, V   
k, W

Proof of Theorem wlkcompim
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlk 23568 . . . 4  |- 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 3081 . . . . 5  |-  v  e. 
_V
3 vex 3081 . . . . 5  |-  e  e. 
_V
4 wlks 23578 . . . . . 6  |-  ( ( 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 6226 . . . . . 6  |-  ( v Walks 
e )  e.  _V
64, 5syl6eqelr 2551 . . . . 5  |-  ( ( 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 . . . 4  |-  { <. 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 5149 . . . . . . . . 9  |-  ( e  =  E  ->  dom  e  =  dom  E )
9 wrdeq 12370 . . . . . . . . 9  |-  ( dom  e  =  dom  E  -> Word 
dom  e  = Word  dom  E )
108, 9syl 16 . . . . . . . 8  |-  ( e  =  E  -> Word  dom  e  = Word  dom  E )
1110eleq2d 2524 . . . . . . 7  |-  ( e  =  E  ->  (
f  e. Word  dom  e  <->  f  e. Word  dom 
E ) )
1211adantl 466 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f  e. Word  dom  e 
<->  f  e. Word  dom  E
) )
13 feq3 5653 . . . . . . 7  |-  ( v  =  V  ->  (
p : ( 0 ... ( # `  f
) ) --> v  <->  p :
( 0 ... ( # `
 f ) ) --> V ) )
1413adantr 465 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( p : ( 0 ... ( # `  f ) ) --> v  <-> 
p : ( 0 ... ( # `  f
) ) --> V ) )
15 fveq1 5799 . . . . . . . . 9  |-  ( e  =  E  ->  (
e `  ( f `  k ) )  =  ( E `  (
f `  k )
) )
1615adantl 466 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e `  (
f `  k )
)  =  ( E `
 ( f `  k ) ) )
1716eqeq1d 2456 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( e `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  (
f `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
1817ralbidv 2846 . . . . . 6  |-  ( ( 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 . . . . 5  |-  ( ( 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 4464 . . . 4  |-  ( ( 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 6418 . . 3  |-  ( 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 elopaelxp 30284 . . . 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 ) ) } ) }  ->  W  e.  ( _V  X.  _V ) )
23 wlkcomp.1 . . . . . 6  |-  F  =  ( 1st `  W
)
24 wlkcomp.2 . . . . . 6  |-  P  =  ( 2nd `  W
)
2523, 24wlkcomp 30435 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  ( _V  X.  _V ) )  ->  ( W  e.  ( V Walks  E )  <->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
2625biimpd 207 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  ( _V  X.  _V ) )  ->  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2722, 26syl3an3 1254 . . 3  |-  ( ( 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 ) ) } ) } )  ->  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
2821, 27sylbi 195 . 2  |-  ( W  e.  ( V Walks  E
)  ->  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
2928pm2.43i 47 1  |-  ( W  e.  ( V Walks  E
)  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078   {cpr 3988   {copab 4458    X. cxp 4947   dom cdm 4949   -->wf 5523   ` cfv 5527  (class class class)co 6201   1stc1st 6686   2ndc2nd 6687   0cc0 9394   1c1 9395    + caddc 9397   ...cfz 11555  ..^cfzo 11666   #chash 12221  Word cword 12340   Walks cwalk 23558
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-wlk 23568
This theorem is referenced by:  usg2wlkeq  30438
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