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Theorem trlon 24218
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 3122 . . . . 5  |-  ( V  e.  X  ->  V  e.  _V )
21ad2antrr 725 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  V  e.  _V )
3 elex 3122 . . . . . 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 22 . . . . . . 7  |-  ( V  e.  X  ->  V  e.  X )
76ancli 551 . . . . . 6  |-  ( V  e.  X  ->  ( V  e.  X  /\  V  e.  X )
)
87ad2antrr 725 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V  e.  X  /\  V  e.  X
) )
9 mpt2exga 6856 . . . . 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 16 . . . 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 6291 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1312oveqd 6299 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1413breqd 4458 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
15 oveq12 6291 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Trails  e )  =  ( V Trails  E
) )
1615breqd 4458 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Trails 
e ) p  <->  f ( V Trails  E ) p ) )
1714, 16anbi12d 710 . . . . . . 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 4510 . . . . . 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 6341 . . . . 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 24191 . . . . 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 6414 . . . 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 1228 . . 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 6299 . 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 756 . . 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 4510 . . . . 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 24217 . . . . . . 7  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
3130wlkres 24198 . . . . . 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 2555 . . . 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 6291 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V WalkOn  E ) b )  =  ( A ( V WalkOn  E ) B ) )
3635breqd 4458 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V WalkOn  E ) b ) p  <->  f ( A ( V WalkOn  E
) B ) p ) )
3736anbi1d 704 . . . . 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 4510 . . . 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 2467 . . . 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 6414 . . 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 1228 . 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 2508 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   class class class wbr 4447   {copab 4504  (class class class)co 6282    |-> cmpt2 6284   Trails ctrail 24175   WalkOn cwlkon 24178   TrailOn ctrlon 24179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-word 12504  df-wlk 24184  df-trail 24185  df-trlon 24191
This theorem is referenced by:  istrlon  24219  trlonprop  24220
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