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Theorem iswlkg 24726
Description: Generalisation of iswlk 24722: 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 24725 . . 3  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
2 iswlk 24722 . . . . 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 1012 . . 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 6298 . . . . . . . . . . 11  |-  ( 0 ... ( # `  F
) )  e.  _V
7 fpmg 7437 . . . . . . . . . . 11  |-  ( ( ( 0 ... ( # `
 F ) )  e.  _V  /\  V  e.  X  /\  P :
( 0 ... ( # `
 F ) ) --> V )  ->  P  e.  ( V  ^pm  (
0 ... ( # `  F
) ) ) )
86, 7mp3an1 1309 . . . . . . . . . 10  |-  ( ( V  e.  X  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P  e.  ( V  ^pm  ( 0 ... ( # `  F
) ) ) )
98ex 432 . . . . . . . . 9  |-  ( V  e.  X  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  P  e.  ( V  ^pm  ( 0 ... ( # `
 F ) ) ) ) )
109anim2d 563 . . . . . . . 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 463 . . . . . . 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 1014 . . . . 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 428 . . . 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 24722 . . . . . . 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 433 . . . . 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 429 . . . 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 432 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   {cpr 4018   class class class wbr 4439   dom cdm 4988   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   0cc0 9481   1c1 9482    + caddc 9484   NN0cn0 10791   ...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:  wlkcomp  24727  trls  24740  constr2wlk  24802  isclwlk  24958
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