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Theorem wlkon 24738
Description: The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
Assertion
Ref Expression
wlkon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V WalkOn  E ) B )  =  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } )
Distinct variable groups:    f, E, p    f, V, p    f, X, p    f, Y, p    A, f, p    B, f, p

Proof of Theorem wlkon
Dummy variables  a 
b  e  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3115 . . . . 5  |-  ( V  e.  X  ->  V  e.  _V )
21ad2antrr 723 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  V  e.  _V )
3 elex 3115 . . . . . 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 723 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V  e.  X  /\  V  e.  X
) )
9 mpt2exga 6849 . . . . 5  |-  ( ( V  e.  X  /\  V  e.  X )  ->  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b ) } )  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 ( V Walks  E ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b ) } )  e.  _V )
11 simpl 455 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
12 oveq12 6279 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
1312breqd 4450 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
14133anbi1d 1301 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b )  <-> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) ) )
1514opabbidv 4502 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  { <. f ,  p >.  |  ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b ) }  =  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } )
1611, 11, 15mpt2eq123dv 6332 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a  e.  v ,  b  e.  v 
|->  { <. f ,  p >.  |  ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b ) } )  =  ( a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) } ) )
17 df-wlkon 24719 . . . . 5  |- WalkOn  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) } ) )
1816, 17ovmpt2ga 6405 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) } )  e.  _V )  ->  ( V WalkOn  E )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } ) )
192, 5, 10, 18syl3anc 1226 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( V WalkOn  E )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } ) )
2019oveqd 6287 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V WalkOn  E ) B )  =  ( A ( a  e.  V , 
b  e.  V  |->  {
<. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  a  /\  ( p `  ( # `
 f ) )  =  b ) } ) B ) )
21 simpl 455 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
2221adantl 464 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
23 simprr 755 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
24 3anass 975 . . . . . . 7  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B )  <->  ( f ( V Walks  E ) p  /\  ( ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  B ) ) )
2524a1i 11 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B )  <-> 
( f ( V Walks 
E ) p  /\  ( ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) ) ) )
2625opabbidv 4502 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) }  =  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  B ) ) } )
27 id 22 . . . . . . 7  |-  ( f ( V Walks  E ) p  ->  f ( V Walks  E ) p )
2827wlkres 24727 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  B ) ) }  e.  _V )
291, 3, 28syl2an 475 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  B ) ) }  e.  _V )
3026, 29eqeltrd 2542 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) }  e.  _V )
3130adantr 463 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) }  e.  _V )
32 eqeq2 2469 . . . . . . 7  |-  ( a  =  A  ->  (
( p `  0
)  =  a  <->  ( p `  0 )  =  A ) )
3332adantr 463 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( p ` 
0 )  =  a  <-> 
( p `  0
)  =  A ) )
34 eqeq2 2469 . . . . . . 7  |-  ( b  =  B  ->  (
( p `  ( # `
 f ) )  =  b  <->  ( p `  ( # `  f
) )  =  B ) )
3534adantl 464 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( p `  ( # `  f ) )  =  b  <->  ( p `  ( # `  f
) )  =  B ) )
3633, 353anbi23d 1300 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b )  <-> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  B ) ) )
3736opabbidv 4502 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b ) }  =  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } )
38 eqid 2454 . . . 4  |-  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } )  =  ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  a  /\  ( p `  ( # `  f ) )  =  b ) } )
3937, 38ovmpt2ga 6405 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  {
<. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  B ) }  e.  _V )  -> 
( A ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } ) B )  =  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } )
4022, 23, 31, 39syl3anc 1226 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( a  e.  V ,  b  e.  V  |->  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  a  /\  (
p `  ( # `  f
) )  =  b ) } ) B )  =  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } )
4120, 40eqtrd 2495 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V WalkOn  E ) B )  =  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   class class class wbr 4439   {copab 4496   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481   #chash 12390   Walks cwalk 24703   WalkOn cwlkon 24707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-wlk 24713  df-wlkon 24719
This theorem is referenced by:  iswlkon  24739  wlkonprop  24740
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