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Theorem iswlk 23424
Description: Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Assertion
Ref Expression
iswlk  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( 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)    W( k)    X( k)    Y( k)    Z( k)

Proof of Theorem iswlk
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4291 . . 3  |-  ( F ( V Walks  E ) P  <->  <. F ,  P >.  e.  ( V Walks  E
) )
2 wlks 23423 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) ) } ) } )
32adantr 465 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( 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 ) ) } ) } )
43eleq2d 2508 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  ( V Walks  E
)  <->  <. F ,  P >.  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 ) ) } ) } ) )
51, 4syl5bb 257 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Walks 
E ) P  <->  <. F ,  P >.  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 ) ) } ) } ) )
6 simpl 457 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  f  =  F )
76eleq1d 2507 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f  e. Word  dom  E  <-> 
F  e. Word  dom  E ) )
8 simpr 461 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
9 fveq2 5689 . . . . . . . 8  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
109oveq2d 6105 . . . . . . 7  |-  ( f  =  F  ->  (
0 ... ( # `  f
) )  =  ( 0 ... ( # `  F ) ) )
1110adantr 465 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 0 ... ( # `
 f ) )  =  ( 0 ... ( # `  F
) ) )
128, 11feq12d 5546 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p : ( 0 ... ( # `  f ) ) --> V  <-> 
P : ( 0 ... ( # `  F
) ) --> V ) )
139adantr 465 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
1413oveq2d 6105 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 0..^ ( # `  f ) )  =  ( 0..^ ( # `  F ) ) )
15 fveq1 5688 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
1615fveq2d 5693 . . . . . . 7  |-  ( f  =  F  ->  ( E `  ( f `  k ) )  =  ( E `  ( F `  k )
) )
17 fveq1 5688 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  k )  =  ( P `  k ) )
18 fveq1 5688 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  ( k  +  1 ) )  =  ( P `  ( k  +  1 ) ) )
1917, 18preq12d 3960 . . . . . . 7  |-  ( p  =  P  ->  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
2016, 19eqeqan12d 2456 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( E `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
2114, 20raleqbidv 2929 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 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 ) ) } ) )
227, 12, 213anbi123d 1289 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( 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 ) ) } ) ) )
2322opelopabga 4600 . . 3  |-  ( ( F  e.  W  /\  P  e.  Z )  ->  ( <. F ,  P >.  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  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2423adantl 466 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  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  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
255, 24bitrd 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( 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 1369    e. wcel 1756   A.wral 2713   {cpr 3877   <.cop 3881   class class class wbr 4290   {copab 4347   dom cdm 4838   -->wf 5412   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281    + caddc 9283   ...cfz 11435  ..^cfzo 11546   #chash 12101  Word cword 12219   Walks cwalk 23403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-word 12227  df-wlk 23413
This theorem is referenced by:  2mwlk  23425  trls  23433  trliswlk  23436  0wlk  23442  wlkntrl  23459  usgrnloop  23460  is2wlk  23462  constr2wlk  23495  redwlk  23503  wlkdvspth  23505  iswlkg  30282  edgwlk  30291  usgra2wlkspth  30295  wlkiswwlk1  30321  wlkiswwlk2  30328
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