MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  el2wlksot Structured version   Visualization version   Unicode version

Theorem el2wlksot 25608
Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
Assertion
Ref Expression
el2wlksot  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( T  e.  ( V 2WalksOt  E )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) ) )
Distinct variable groups:    T, a,
b, c, f, p    E, a, b, c, f, p    V, a, b, c, f, p    X, a, b, c, f, p    Y, a, b, c, f, p

Proof of Theorem el2wlksot
StepHypRef Expression
1 el2wlksoton 25606 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( T  e.  ( V 2WalksOt  E )  <->  E. a  e.  V  E. c  e.  V  T  e.  ( a ( V 2WalksOnOt  E ) c ) ) )
2 el2wlkonot 25597 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( a  e.  V  /\  c  e.  V ) )  -> 
( T  e.  ( a ( V 2WalksOnOt  E ) c )  <->  E. b  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) ) )
322rexbidva 2907 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E. a  e.  V  E. c  e.  V  T  e.  ( a ( V 2WalksOnOt  E ) c )  <->  E. a  e.  V  E. c  e.  V  E. b  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) ) )
4 rexcom 2952 . . . 4  |-  ( E. c  e.  V  E. b  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  (
a  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  c  =  ( p `  2 ) ) ) )  <->  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) )
54a1i 11 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E. c  e.  V  E. b  e.  V  ( T  = 
<. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( a  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  c  =  ( p `  2 ) ) ) )  <->  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) ) )
65rexbidv 2901 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( E. a  e.  V  E. c  e.  V  E. b  e.  V  ( T  = 
<. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( a  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  c  =  ( p `  2 ) ) ) )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) ) )
71, 3, 63bitrd 283 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( T  e.  ( V 2WalksOt  E )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( a  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  c  =  ( p `  2
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   E.wrex 2738   <.cotp 3976   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540   2c2 10659   #chash 12515   Walks cwalk 25226   2WalksOt c2wlkot 25582   2WalksOnOt c2wlkonot 25583
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-ot 3977  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-wlkon 25242  df-2wlkonot 25586  df-2wlksot 25587
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator