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Theorem pthon 23606
Description: The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
Assertion
Ref Expression
pthon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V PathOn  E ) B )  =  { <. f ,  p >.  |  (
f ( A ( V WalkOn  E ) B ) p  /\  f
( V Paths  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 pthon
Dummy variables  a 
b  e  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3074 . . . . 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 3074 . . . . . 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 6746 . . . . 5  |-  ( ( V  e.  X  /\  V  e.  X )  ->  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Paths 
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 Paths 
E ) p ) } )  e.  _V )
11 simpl 457 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
12 oveq12 6196 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1312oveqd 6204 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1413breqd 4398 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
15 oveq12 6196 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Paths  e )  =  ( V Paths  E
) )
1615breqd 4398 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Paths 
e ) p  <->  f ( V Paths  E ) p ) )
1714, 16anbi12d 710 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Paths 
e ) p )  <-> 
( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Paths  E
) p ) ) )
1817opabbidv 4450 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Paths 
e ) p ) }  =  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Paths  E )
p ) } )
1911, 11, 18mpt2eq123dv 6244 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a  e.  v ,  b  e.  v 
|->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Paths 
e ) p ) } )  =  ( a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Paths  E
) p ) } ) )
20 df-pthon 23555 . . . . 5  |- PathOn  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  {
<. f ,  p >.  |  ( f ( a ( v WalkOn  e ) b ) p  /\  f ( v Paths  e
) p ) } ) )
2119, 20ovmpt2ga 6317 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Paths  E
) p ) } )  e.  _V )  ->  ( V PathOn  E )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( a ( V WalkOn  E ) b ) p  /\  f
( V Paths  E )
p ) } ) )
222, 5, 10, 21syl3anc 1219 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V PathOn  E )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( a ( V WalkOn  E ) b ) p  /\  f
( V Paths  E )
p ) } ) )
2322oveqd 6204 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V PathOn  E ) B )  =  ( A ( a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Paths  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 Paths  E )
p )  <->  ( f
( V Paths  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 Paths 
E ) p )  <-> 
( f ( V Paths 
E ) p  /\  f ( A ( V WalkOn  E ) B ) p ) ) )
2928opabbidv 4450 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Paths 
E ) p ) }  =  { <. f ,  p >.  |  ( f ( V Paths  E
) p  /\  f
( A ( V WalkOn  E ) B ) p ) } )
30 pthistrl 23603 . . . . . . . 8  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
31 trliswlk 23570 . . . . . . . 8  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
3230, 31syl 16 . . . . . . 7  |-  ( f ( V Paths  E ) p  ->  f ( V Walks  E ) p )
3332wlkres 23560 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Paths  E ) p  /\  f ( A ( V WalkOn  E ) B ) p ) }  e.  _V )
341, 3, 33syl2an 477 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Paths  E ) p  /\  f ( A ( V WalkOn  E ) B ) p ) }  e.  _V )
3529, 34eqeltrd 2537 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Paths 
E ) p ) }  e.  _V )
3635adantr 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 Paths 
E ) p ) }  e.  _V )
37 oveq12 6196 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V WalkOn  E ) b )  =  ( A ( V WalkOn  E ) B ) )
3837breqd 4398 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V WalkOn  E ) b ) p  <->  f ( A ( V WalkOn  E
) B ) p ) )
3938anbi1d 704 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Paths 
E ) p )  <-> 
( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Paths  E
) p ) ) )
4039opabbidv 4450 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E
) b ) p  /\  f ( V Paths 
E ) p ) }  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f
( V Paths  E )
p ) } )
41 eqid 2451 . . . 4  |-  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Paths  E )
p ) } )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( a ( V WalkOn  E ) b ) p  /\  f
( V Paths  E )
p ) } )
4240, 41ovmpt2ga 6317 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  {
<. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Paths  E
) p ) }  e.  _V )  -> 
( A ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( a ( V WalkOn  E ) b ) p  /\  f
( V Paths  E )
p ) } ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Paths  E
) p ) } )
4325, 26, 36, 42syl3anc 1219 . 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 Paths  E )
p ) } ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Paths  E
) p ) } )
4423, 43eqtrd 2491 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V PathOn  E ) B )  =  { <. f ,  p >.  |  (
f ( A ( V WalkOn  E ) B ) p  /\  f
( V Paths  E )
p ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065   class class class wbr 4387   {copab 4444  (class class class)co 6187    |-> cmpt2 6189   Walks cwalk 23537   Trails ctrail 23538   Paths cpath 23539   WalkOn cwlkon 23541   PathOn cpthon 23543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-map 7313  df-pm 7314  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-word 12328  df-wlk 23547  df-trail 23548  df-pth 23549  df-pthon 23555
This theorem is referenced by:  ispthon  23607  pthonprop  23608
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