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Theorem istrl 23455
Description: Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Assertion
Ref Expression
istrl  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Trails  E ) P  <->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  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 istrl
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4312 . . 3  |-  ( F ( V Trails  E ) P  <->  <. F ,  P >.  e.  ( V Trails  E
) )
2 trls 23454 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Trails  E )  =  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  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 Trails  E )  =  { <. f ,  p >.  |  ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) } )
43eleq2d 2510 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  ( V Trails  E
)  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  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 Trails  E ) P  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } ) )
6 eleq1 2503 . . . . . . 7  |-  ( f  =  F  ->  (
f  e. Word  dom  E  <->  F  e. Word  dom 
E ) )
7 cnveq 5032 . . . . . . . 8  |-  ( f  =  F  ->  `' f  =  `' F
)
87funeqd 5458 . . . . . . 7  |-  ( f  =  F  ->  ( Fun  `' f  <->  Fun  `' F ) )
96, 8anbi12d 710 . . . . . 6  |-  ( f  =  F  ->  (
( f  e. Word  dom  E  /\  Fun  `' f )  <->  ( F  e. Word  dom  E  /\  Fun  `' F ) ) )
109adantr 465 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f  e. Word  dom  E  /\  Fun  `' f )  <->  ( F  e. Word  dom  E  /\  Fun  `' F ) ) )
11 simpr 461 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
12 fveq2 5710 . . . . . . . 8  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
1312oveq2d 6126 . . . . . . 7  |-  ( f  =  F  ->  (
0 ... ( # `  f
) )  =  ( 0 ... ( # `  F ) ) )
1413adantr 465 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 0 ... ( # `
 f ) )  =  ( 0 ... ( # `  F
) ) )
1511, 14feq12d 5567 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p : ( 0 ... ( # `  f ) ) --> V  <-> 
P : ( 0 ... ( # `  F
) ) --> V ) )
1612oveq2d 6126 . . . . . . 7  |-  ( f  =  F  ->  (
0..^ ( # `  f
) )  =  ( 0..^ ( # `  F
) ) )
1716adantr 465 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 0..^ ( # `  f ) )  =  ( 0..^ ( # `  F ) ) )
18 fveq1 5709 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
1918fveq2d 5714 . . . . . . 7  |-  ( f  =  F  ->  ( E `  ( f `  k ) )  =  ( E `  ( F `  k )
) )
20 fveq1 5709 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  k )  =  ( P `  k ) )
21 fveq1 5709 . . . . . . . 8  |-  ( p  =  P  ->  (
p `  ( k  +  1 ) )  =  ( P `  ( k  +  1 ) ) )
2220, 21preq12d 3981 . . . . . . 7  |-  ( p  =  P  ->  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
2319, 22eqeqan12d 2458 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( E `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
2417, 23raleqbidv 2950 . . . . 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 ) ) } ) )
2510, 15, 243anbi123d 1289 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  <->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2625opelopabga 4621 . . 3  |-  ( ( F  e.  W  /\  P  e.  Z )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) }  <-> 
( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2726adantl 466 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) }  <-> 
( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
285, 27bitrd 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Trails  E ) P  <->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  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 2734   {cpr 3898   <.cop 3902   class class class wbr 4311   {copab 4368   `'ccnv 4858   dom cdm 4859   Fun wfun 5431   -->wf 5433   ` cfv 5437  (class class class)co 6110   0cc0 9301   1c1 9302    + caddc 9304   ...cfz 11456  ..^cfzo 11567   #chash 12122  Word cword 12240   Trails ctrail 23425
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 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-map 7235  df-pm 7236  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-n0 10599  df-z 10666  df-uz 10881  df-fz 11457  df-fzo 11568  df-word 12248  df-wlk 23434  df-trail 23435
This theorem is referenced by:  istrl2  23456  trliswlk  23457  0trl  23464  wlkntrl  23480  wlkdvspth  23526  3v3e3cycl1  23549  constr3trl  23564  4cycl4v4e  23571  4cycl4dv4e  23573  eupatrl  23608  usgra2wlkspth  30321
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