MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkon Structured version   Unicode version

Theorem wlkon 24237
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 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 6859 . . . . 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 457 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
12 oveq12 6293 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
1312breqd 4458 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
14133anbi1d 1303 . . . . . . 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 4510 . . . . . 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 6343 . . . . 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 24218 . . . . 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 6416 . . . 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 1228 . . 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 6301 . 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 457 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
2221adantl 466 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
23 simprr 756 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
24 3anass 977 . . . . . . 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 4510 . . . . 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 24226 . . . . . 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 477 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  B ) ) }  e.  _V )
3026, 29eqeltrd 2555 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) }  e.  _V )
3130adantr 465 . . 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 2482 . . . . . . 7  |-  ( a  =  A  ->  (
( p `  0
)  =  a  <->  ( p `  0 )  =  A ) )
3332adantr 465 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( p ` 
0 )  =  a  <-> 
( p `  0
)  =  A ) )
34 eqeq2 2482 . . . . . . 7  |-  ( b  =  B  ->  (
( p `  ( # `
 f ) )  =  b  <->  ( p `  ( # `  f
) )  =  B ) )
3534adantl 466 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( p `  ( # `  f ) )  =  b  <->  ( p `  ( # `  f
) )  =  B ) )
3633, 353anbi23d 1302 . . . . 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 4510 . . . 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 2467 . . . 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 6416 . . 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 1228 . 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 2508 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   class class class wbr 4447   {copab 4504   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   0cc0 9492   #chash 12373   Walks cwalk 24202   WalkOn cwlkon 24206
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-word 12508  df-wlk 24212  df-wlkon 24218
This theorem is referenced by:  iswlkon  24238  wlkonprop  24239
  Copyright terms: Public domain W3C validator