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Theorem 2pthwlkonot 30416
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 994 . . . . . . . . . . . 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 996 . . . . . . . . . . . 12  |-  ( ( ( # `  f
)  =  2  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B ) )  ->  A  =/=  B )
4 usgra2wlkspth 30310 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 969 . . . . . 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 969 . . . . . 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 1682 . . . 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 2975 . 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 23282 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
18 2spthonot 30397 . . . 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 1008 . 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 30396 . . . 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 1008 . 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 2485 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2618   {crab 2731   _Vcvv 2984   class class class wbr 4304    X. cxp 4850   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588   1c1 9295   2c2 10383   #chash 12115   USGrph cusg 23276   WalkOn cwlkon 23421   SPathOn cspthon 23424   2WalksOnOt c2wlkonot 30386   2SPathOnOt c2pthonot 30388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-hash 12116  df-word 12241  df-usgra 23278  df-wlk 23427  df-trail 23428  df-pth 23429  df-spth 23430  df-wlkon 23433  df-spthon 23436  df-2wlkonot 30389  df-2spthonot 30391
This theorem is referenced by:  usg2spthonot  30419  usg2spthonot0  30420  frg2spot1  30663
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