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Theorem iswlkg 30434
Description: Generalisation of iswlk 23579: Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Assertion
Ref Expression
iswlkg  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V Walks 
E ) P  <->  ( 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
Allowed substitution hints:    V( k)    X( k)    Y( k)

Proof of Theorem iswlkg
StepHypRef Expression
1 wlkbprop 23586 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
2 iswlk 23579 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P 
<->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
32biimpd 207 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
433adant1 1006 . . 3  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
51, 4mpcom 36 . 2  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
6 ovex 6226 . . . . . . . . . . 11  |-  ( 0 ... ( # `  F
) )  e.  _V
7 fpmg 7349 . . . . . . . . . . 11  |-  ( ( ( 0 ... ( # `
 F ) )  e.  _V  /\  V  e.  X  /\  P :
( 0 ... ( # `
 F ) ) --> V )  ->  P  e.  ( V  ^pm  (
0 ... ( # `  F
) ) ) )
86, 7mp3an1 1302 . . . . . . . . . 10  |-  ( ( V  e.  X  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... ( # `  F
) ) ) )
98ex 434 . . . . . . . . 9  |-  ( V  e.  X  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  P  e.  ( V  ^pm  ( 0 ... ( # `
 F ) ) ) ) )
109anim2d 565 . . . . . . . 8  |-  ( V  e.  X  ->  (
( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V )  ->  ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  (
0 ... ( # `  F
) ) ) ) ) )
1110adantr 465 . . . . . . 7  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V )  ->  ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  ( 0 ... ( # `
 F ) ) ) ) ) )
1211com12 31 . . . . . 6  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  ( 0 ... ( # `
 F ) ) ) ) ) )
13123adant3 1008 . . . . 5  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  (
( V  e.  X  /\  E  e.  Y
)  ->  ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  (
0 ... ( # `  F
) ) ) ) ) )
1413impcom 430 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( 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  e.  ( V  ^pm  ( 0 ... ( # `  F
) ) ) ) )
15 iswlk 23579 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  (
0 ... ( # `  F
) ) ) ) )  ->  ( F
( V Walks  E ) P 
<->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
1615biimprd 223 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  (
0 ... ( # `  F
) ) ) ) )  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  F
( V Walks  E ) P ) )
1716expcom 435 . . . . 5  |-  ( ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  ( 0 ... ( # `  F
) ) ) )  ->  ( ( V  e.  X  /\  E  e.  Y )  ->  (
( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  ->  F ( V Walks  E
) P ) ) )
1817impd 431 . . . 4  |-  ( ( F  e. Word  dom  E  /\  P  e.  ( V  ^pm  ( 0 ... ( # `  F
) ) ) )  ->  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) )  ->  F ( V Walks 
E ) P ) )
1914, 18mpcom 36 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )  ->  F ( V Walks  E
) P )
2019ex 434 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  ->  F ( V Walks  E
) P ) )
215, 20impbid2 204 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V Walks 
E ) P  <->  ( 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   class class class wbr 4401   dom cdm 4949   -->wf 5523   ` cfv 5527  (class class class)co 6201    ^pm cpm 7326   0cc0 9394   1c1 9395    + caddc 9397   NN0cn0 10691   ...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:  wlkcomp  30435  isclwlk  30570
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