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Theorem pthon 25305
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 3054 . . . . 5  |-  ( V  e.  X  ->  V  e.  _V )
21ad2antrr 732 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  V  e.  _V )
3 elex 3054 . . . . . 6  |-  ( E  e.  Y  ->  E  e.  _V )
43adantl 468 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
54adantr 467 . . . 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 554 . . . . . 6  |-  ( V  e.  X  ->  ( V  e.  X  /\  V  e.  X )
)
87ad2antrr 732 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V  e.  X  /\  V  e.  X
) )
9 mpt2exga 6869 . . . . 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 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 Paths 
E ) p ) } )  e.  _V )
11 simpl 459 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
12 oveq12 6299 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1312oveqd 6307 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1413breqd 4413 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
15 oveq12 6299 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Paths  e )  =  ( V Paths  E
) )
1615breqd 4413 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Paths 
e ) p  <->  f ( V Paths  E ) p ) )
1714, 16anbi12d 717 . . . . . . 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 4466 . . . . . 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 6353 . . . . 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 25244 . . . . 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 6426 . . . 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 1268 . . 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 6307 . 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 459 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
2524adantl 468 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
26 simprr 766 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
27 ancom 452 . . . . . . 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 4466 . . . . 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 25302 . . . . . . . 8  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
31 trliswlk 25269 . . . . . . . 8  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
3230, 31syl 17 . . . . . . 7  |-  ( f ( V Paths  E ) p  ->  f ( V Walks  E ) p )
3332wlkres 25250 . . . . . 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 480 . . . . 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 2529 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Paths 
E ) p ) }  e.  _V )
3635adantr 467 . . 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 6299 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V WalkOn  E ) b )  =  ( A ( V WalkOn  E ) B ) )
3837breqd 4413 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V WalkOn  E ) b ) p  <->  f ( A ( V WalkOn  E
) B ) p ) )
3938anbi1d 711 . . . . 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 4466 . . . 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 6426 . . 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 1268 . 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 2485 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045   class class class wbr 4402   {copab 4460  (class class class)co 6290    |-> cmpt2 6292   Walks cwalk 25226   Trails ctrail 25227   Paths cpath 25228   WalkOn cwlkon 25230   PathOn cpthon 25232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-wlk 25236  df-trail 25237  df-pth 25238  df-pthon 25244
This theorem is referenced by:  ispthon  25306  pthonprop  25307
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