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Theorem trls 25111
Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Assertion
Ref Expression
trls  |-  ( ( 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 ) ) } ) } )
Distinct variable groups:    f, E, k, p    f, V, p
Allowed substitution hints:    V( k)    X( f, k, p)    Y( f,
k, p)

Proof of Theorem trls
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3096 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3096 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-trail 25082 . . . 4  |- Trails  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  Fun  `' f ) } )
43a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> Trails 
=  ( v  e. 
_V ,  e  e. 
_V  |->  { <. f ,  p >.  |  (
f ( v Walks  e
) p  /\  Fun  `' f ) } ) )
5 oveq12 6314 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
65breqd 4437 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
76anbi1d 709 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  Fun  `' f )  <->  ( f ( V Walks  E ) p  /\  Fun  `' f ) ) )
87adantl 467 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v Walks 
e ) p  /\  Fun  `' f )  <->  ( f
( V Walks  E )
p  /\  Fun  `' f ) ) )
9 iswlkg 25097 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  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 ) ) } ) ) )
109adantr 466 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
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 ) ) } ) ) )
1110anbi1d 709 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( V Walks 
E ) p  /\  Fun  `' f )  <->  ( (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) ) )
12 3anass 986 . . . . . . 7  |-  ( ( ( 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 ) ) } ) ) )
13 ancom 451 . . . . . . . 8  |-  ( ( f  e. Word  dom  E  /\  Fun  `' f )  <-> 
( Fun  `' f  /\  f  e. Word  dom  E
) )
1413anbi1i 699 . . . . . . 7  |-  ( ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )  <->  ( ( Fun  `' f  /\  f  e. Word  dom  E )  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) )
15 anass 653 . . . . . . . 8  |-  ( ( ( Fun  `' f  /\  f  e. Word  dom  E )  /\  ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )  <->  ( Fun  `' f  /\  ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) ) )
16 ancom 451 . . . . . . . 8  |-  ( ( Fun  `' f  /\  ( 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 ) ) } ) )  /\  Fun  `' f ) )
17 3anass 986 . . . . . . . . . 10  |-  ( ( 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 ) ) } ) ) )
1817bicomi 205 . . . . . . . . 9  |-  ( ( 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 ) ) } ) )
1918anbi1i 699 . . . . . . . 8  |-  ( ( ( f  e. Word  dom  E  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )  /\  Fun  `' f )  <->  ( ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) )
2015, 16, 193bitri 274 . . . . . . 7  |-  ( ( ( Fun  `' f  /\  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 ) ) } )  /\  Fun  `' f ) )
2112, 14, 203bitri 274 . . . . . 6  |-  ( ( ( 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  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  /\  Fun  `' f ) )
2211, 21syl6bbr 266 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( V Walks 
E ) p  /\  Fun  `' f )  <->  ( (
f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
238, 22bitrd 256 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v Walks 
e ) p  /\  Fun  `' f )  <->  ( (
f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
2423opabbidv 4489 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  Fun  `' f ) }  =  { <. 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 ) ) } ) } )
25 simpl 458 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
26 simpr 462 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
27 anass 653 . . . . . 6  |-  ( ( ( 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 ) ) } ) ) ) )
2812, 27bitri 252 . . . . 5  |-  ( ( ( 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 ) ) } ) ) ) )
2928opabbii 4490 . . . 4  |-  { <. 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 ,  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 ) ) } ) ) ) }
30 dmexg 6738 . . . . . . 7  |-  ( E  e.  _V  ->  dom  E  e.  _V )
3130adantl 467 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  dom  E  e.  _V )
32 wrdexg 12669 . . . . . 6  |-  ( dom 
E  e.  _V  -> Word  dom  E  e.  _V )
3331, 32syl 17 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> Word 
dom  E  e.  _V )
34 fzfi 12182 . . . . . . 7  |-  ( 0 ... ( # `  f
) )  e.  Fin
3525adantr 466 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  V  e.  _V )
36 mapex 7486 . . . . . . 7  |-  ( ( ( 0 ... ( # `
 f ) )  e.  Fin  /\  V  e.  _V )  ->  { p  |  p : ( 0 ... ( # `  f
) ) --> V }  e.  _V )
3734, 35, 36sylancr 667 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  { p  |  p : ( 0 ... ( # `  f
) ) --> V }  e.  _V )
38 simpl 458 . . . . . . . . 9  |-  ( ( p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  ->  p : ( 0 ... ( # `  f
) ) --> V )
3938adantl 467 . . . . . . . 8  |-  ( ( Fun  `' f  /\  ( p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )  ->  p : ( 0 ... ( # `  f ) ) --> V )
4039ss2abi 3539 . . . . . . 7  |-  { p  |  ( Fun  `' f  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) }  C_  { p  |  p : ( 0 ... ( # `  f
) ) --> V }
4140a1i 11 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  { p  |  ( Fun  `' f  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) }  C_  { p  |  p : ( 0 ... ( # `  f
) ) --> V }
)
4237, 41ssexd 4572 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  f  e. Word  dom  E
)  ->  { p  |  ( Fun  `' f  /\  ( p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) ) }  e.  _V )
4333, 42opabex3d 6785 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. 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 ) ) } ) ) ) }  e.  _V )
4429, 43syl5eqel 2521 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. 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 ) ) } ) }  e.  _V )
454, 24, 25, 26, 44ovmpt2d 6438 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( 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 ) ) } ) } )
461, 2, 45syl2an 479 1  |-  ( ( 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 ) ) } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {cab 2414   A.wral 2782   _Vcvv 3087    C_ wss 3442   {cpr 4004   class class class wbr 4426   {copab 4483   `'ccnv 4853   dom cdm 4854   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   Fincfn 7577   0cc0 9538   1c1 9539    + caddc 9541   ...cfz 11782  ..^cfzo 11913   #chash 12512  Word cword 12643   Walks cwalk 25071   Trails ctrail 25072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-wlk 25081  df-trail 25082
This theorem is referenced by:  istrl  25112
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