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Theorem wlkonprop 23568
Description: Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
Assertion
Ref Expression
wlkonprop  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) ) )

Proof of Theorem wlkonprop
Dummy variables  a 
b  e  f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlkon 23558 . . 3  |- 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 ) } ) )
2 oveq12 6201 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
32breqd 4403 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
433anbi1d 1294 . . 3  |-  ( ( 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 ) ) )
5 wlkon 23566 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( A ( V WalkOn  E
) B )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) } )
61, 4, 5bropopvvv 6755 . 2  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
) )
7 iswlkon 23567 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
87biimpd 207 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V WalkOn  E ) B ) P  ->  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) ) )
98imdistani 690 . 2  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  F ( A ( V WalkOn  E
) B ) P )  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) ) )
106, 9mpancom 669 1  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3070   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   0cc0 9385   #chash 12206   Walks cwalk 23542   WalkOn cwlkon 23546
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
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 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-word 12333  df-wlk 23552  df-wlkon 23558
This theorem is referenced by:  wlkoniswlk  23569  usgra2wlkspth  30438
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