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Theorem 2pthwlkonot 24658
Description: For two different vertices, a walk of length 2 between these vertices as ordered triple is a simple path of length 2 between these vertices as ordered triple in an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
2pthwlkonot  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  ( A ( V 2WalksOnOt  E ) B ) )

Proof of Theorem 2pthwlkonot
Dummy variables  f  p  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr1 1002 . . . . . . . . . . . 12  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  V USGrph  E )
2 simpl 457 . . . . . . . . . . . 12  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  ( # `
 f )  =  2 )
3 simpr3 1004 . . . . . . . . . . . 12  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  A  =/=  B )
4 usgra2wlkspth 24394 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( # `
 f )  =  2  /\  A  =/= 
B )  ->  (
f ( A ( V WalkOn  E ) B ) p  <->  f ( A ( V SPathOn  E
) B ) p ) )
51, 2, 3, 4syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  (
f ( A ( V WalkOn  E ) B ) p  <->  f ( A ( V SPathOn  E
) B ) p ) )
65bicomd 201 . . . . . . . . . 10  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  (
f ( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) )
76ex 434 . . . . . . . . 9  |-  ( (
# `  f )  =  2  ->  (
( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( f
( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) ) )
87adantr 465 . . . . . . . 8  |-  ( ( ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) )  ->  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  ->  (
f ( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) ) )
98com12 31 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( (
( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) )  ->  ( f
( A ( V SPathOn  E ) B ) p  <->  f ( A ( V WalkOn  E ) B ) p ) ) )
109pm5.32rd 640 . . . . . 6  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( (
f ( A ( V SPathOn  E ) B ) p  /\  ( (
# `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) )  <->  ( f
( A ( V WalkOn  E ) B ) p  /\  ( (
# `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) ) )
11 3anass 977 . . . . . 6  |-  ( ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  ( f
( A ( V SPathOn  E ) B ) p  /\  ( (
# `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
12 3anass 977 . . . . . 6  |-  ( ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `
 f )  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) )  <->  ( f ( A ( V WalkOn  E
) B ) p  /\  ( ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) ) )
1310, 11, 123bitr4g 288 . . . . 5  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( (
f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  ( f
( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) ) )
14132exbidv 1692 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  E. f E. p ( f ( A ( V WalkOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
1514adantr 465 . . 3  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  /\  t  e.  ( ( V  X.  V )  X.  V
) )  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) )  <->  E. f E. p ( f ( A ( V WalkOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) ) )
1615rabbidva 3104 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
17 usgrav 24111 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
18 2spthonot 24639 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( A ( V 2SPathOnOt  E ) B )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
1917, 18sylan 471 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
20193adant3 1016 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
21 2wlkonot 24638 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( A ( V 2WalksOnOt  E ) B )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
2217, 21sylan 471 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( A
( V 2WalksOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
23223adant3 1016 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2WalksOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) } )
2416, 20, 233eqtr4d 2518 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A
( V 2SPathOnOt  E ) B )  =  ( A ( V 2WalksOnOt  E ) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113   class class class wbr 4447    X. cxp 4997   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   1c1 9494   2c2 10586   #chash 12374   USGrph cusg 24103   WalkOn cwlkon 24275   SPathOn cspthon 24278   2WalksOnOt c2wlkonot 24628   2SPathOnOt c2pthonot 24630
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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-rmo 2822  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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-usgra 24106  df-wlk 24281  df-trail 24282  df-pth 24283  df-spth 24284  df-wlkon 24287  df-spthon 24290  df-2wlkonot 24631  df-2spthonot 24633
This theorem is referenced by:  usg2spthonot  24661  usg2spthonot0  24662  frg2spot1  24832
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