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Theorem pthon 24875
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 3067 . . . . 5  |-  ( V  e.  X  ->  V  e.  _V )
21ad2antrr 724 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  V  e.  _V )
3 elex 3067 . . . . . 6  |-  ( E  e.  Y  ->  E  e.  _V )
43adantl 464 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
54adantr 463 . . . 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 549 . . . . . 6  |-  ( V  e.  X  ->  ( V  e.  X  /\  V  e.  X )
)
87ad2antrr 724 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V  e.  X  /\  V  e.  X
) )
9 mpt2exga 6814 . . . . 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 455 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
12 oveq12 6243 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1312oveqd 6251 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1413breqd 4405 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
15 oveq12 6243 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Paths  e )  =  ( V Paths  E
) )
1615breqd 4405 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Paths 
e ) p  <->  f ( V Paths  E ) p ) )
1714, 16anbi12d 709 . . . . . . 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 4457 . . . . . 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 6296 . . . . 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 24814 . . . . 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 6369 . . . 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 1230 . . 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 6251 . 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 455 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
2524adantl 464 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
26 simprr 758 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
27 ancom 448 . . . . . . 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 4457 . . . . 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 24872 . . . . . . . 8  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
31 trliswlk 24839 . . . . . . . 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 24820 . . . . . 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 475 . . . . 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 2490 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Paths 
E ) p ) }  e.  _V )
3635adantr 463 . . 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 6243 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V WalkOn  E ) b )  =  ( A ( V WalkOn  E ) B ) )
3837breqd 4405 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V WalkOn  E ) b ) p  <->  f ( A ( V WalkOn  E
) B ) p ) )
3938anbi1d 703 . . . . 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 4457 . . . 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 2402 . . . 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 6369 . . 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 1230 . 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 2443 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   class class class wbr 4394   {copab 4451  (class class class)co 6234    |-> cmpt2 6236   Walks cwalk 24796   Trails ctrail 24797   Paths cpath 24798   WalkOn cwlkon 24800   PathOn cpthon 24802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-fzo 11768  df-hash 12360  df-word 12498  df-wlk 24806  df-trail 24807  df-pth 24808  df-pthon 24814
This theorem is referenced by:  ispthon  24876  pthonprop  24877
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