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Theorem trlon 24971
Description: The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
Assertion
Ref Expression
trlon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V TrailOn  E ) B )  =  { <. f ,  p >.  |  (
f ( A ( V WalkOn  E ) B ) p  /\  f
( V Trails  E )
p ) } )
Distinct variable groups:    f, E, p    f, V, p    f, X, p    f, Y, p    A, f, p    B, f, p

Proof of Theorem trlon
Dummy variables  a 
b  e  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3070 . . . . 5  |-  ( V  e.  X  ->  V  e.  _V )
21ad2antrr 726 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  V  e.  _V )
3 elex 3070 . . . . . 6  |-  ( E  e.  Y  ->  E  e.  _V )
43adantl 466 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
54adantr 465 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  E  e.  _V )
6 id 23 . . . . . . 7  |-  ( V  e.  X  ->  V  e.  X )
76ancli 551 . . . . . 6  |-  ( V  e.  X  ->  ( V  e.  X  /\  V  e.  X )
)
87ad2antrr 726 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V  e.  X  /\  V  e.  X
) )
9 mpt2exga 6862 . . . . 5  |-  ( ( V  e.  X  /\  V  e.  X )  ->  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Trails  E ) p ) } )  e.  _V )
108, 9syl 17 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Trails  E ) p ) } )  e.  _V )
11 simpl 457 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
12 oveq12 6289 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1312oveqd 6297 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1413breqd 4408 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
15 oveq12 6289 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Trails  e )  =  ( V Trails  E
) )
1615breqd 4408 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Trails 
e ) p  <->  f ( V Trails  E ) p ) )
1714, 16anbi12d 711 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Trails 
e ) p )  <-> 
( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Trails  E
) p ) ) )
1817opabbidv 4460 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Trails 
e ) p ) }  =  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } )
1911, 11, 18mpt2eq123dv 6342 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a  e.  v ,  b  e.  v 
|->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Trails 
e ) p ) } )  =  ( a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Trails  E
) p ) } ) )
20 df-trlon 24944 . . . . 5  |- TrailOn  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  {
<. f ,  p >.  |  ( f ( a ( v WalkOn  e ) b ) p  /\  f ( v Trails  e
) p ) } ) )
2119, 20ovmpt2ga 6415 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Trails  E
) p ) } )  e.  _V )  ->  ( V TrailOn  E )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } ) )
222, 5, 10, 21syl3anc 1232 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V TrailOn  E )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } ) )
2322oveqd 6297 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V TrailOn  E ) B )  =  ( A ( a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Trails  E
) p ) } ) B ) )
24 simpl 457 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
2524adantl 466 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
26 simprr 760 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
27 ancom 450 . . . . . . 7  |-  ( ( f ( A ( V WalkOn  E ) B ) p  /\  f
( V Trails  E )
p )  <->  ( f
( V Trails  E )
p  /\  f ( A ( V WalkOn  E
) B ) p ) )
2827a1i 11 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Trails  E ) p )  <-> 
( f ( V Trails  E ) p  /\  f ( A ( V WalkOn  E ) B ) p ) ) )
2928opabbidv 4460 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Trails  E ) p ) }  =  { <. f ,  p >.  |  ( f ( V Trails  E
) p  /\  f
( A ( V WalkOn  E ) B ) p ) } )
30 trliswlk 24970 . . . . . . 7  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
3130wlkres 24951 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  f ( A ( V WalkOn  E ) B ) p ) }  e.  _V )
321, 3, 31syl2an 477 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  f ( A ( V WalkOn  E ) B ) p ) }  e.  _V )
3329, 32eqeltrd 2492 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Trails  E ) p ) }  e.  _V )
3433adantr 465 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Trails  E ) p ) }  e.  _V )
35 oveq12 6289 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V WalkOn  E ) b )  =  ( A ( V WalkOn  E ) B ) )
3635breqd 4408 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V WalkOn  E ) b ) p  <->  f ( A ( V WalkOn  E
) B ) p ) )
3736anbi1d 705 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Trails  E ) p )  <-> 
( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E
) p ) ) )
3837opabbidv 4460 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Trails  E ) p ) }  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f
( V Trails  E )
p ) } )
39 eqid 2404 . . . 4  |-  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } )
4038, 39ovmpt2ga 6415 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  {
<. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E
) p ) }  e.  _V )  -> 
( A ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E
) p ) } )
4125, 26, 34, 40syl3anc 1232 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Trails  E )
p ) } ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E
) p ) } )
4223, 41eqtrd 2445 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V TrailOn  E ) B )  =  { <. f ,  p >.  |  (
f ( A ( V WalkOn  E ) B ) p  /\  f
( V Trails  E )
p ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   _Vcvv 3061   class class class wbr 4397   {copab 4454  (class class class)co 6280    |-> cmpt2 6282   Trails ctrail 24928   WalkOn cwlkon 24931   TrailOn ctrlon 24932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-wlk 24937  df-trail 24938  df-trlon 24944
This theorem is referenced by:  istrlon  24972  trlonprop  24973
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