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Theorem constr2spth 23652
Description: A simple path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Hypotheses
Ref Expression
2trlY.i  |-  ( I  e.  U  /\  J  e.  W )
2trlY.f  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
2trlY.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
constr2spth  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( I  =/= 
J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F ( V SPaths  E
) P ) )

Proof of Theorem constr2spth
StepHypRef Expression
1 2trlY.i . . . . . 6  |-  ( I  e.  U  /\  J  e.  W )
2 2trlY.f . . . . . 6  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
3 2trlY.p . . . . . 6  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
41, 2, 3constr2trl 23651 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( I  =/= 
J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F ( V Trails  E
) P ) )
543adant3 1008 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( I  =/= 
J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F ( V Trails  E
) P ) )
65imp 429 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  F ( V Trails  E
) P )
73constr2spthlem1 23646 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  Fun  `' P )
873adant1 1006 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  ->  Fun  `' P )
98adantr 465 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  Fun  `' P )
10 prex 4643 . . . . . . . . 9  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
112, 10eqeltri 2538 . . . . . . . 8  |-  F  e. 
_V
12 tpex 6490 . . . . . . . . 9  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
133, 12eqeltri 2538 . . . . . . . 8  |-  P  e. 
_V
1411, 13pm3.2i 455 . . . . . . 7  |-  ( F  e.  _V  /\  P  e.  _V )
1514jctr 542 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
16153ad2ant1 1009 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
1716adantr 465 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
18 isspth 23621 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  _V  /\  P  e. 
_V ) )  -> 
( F ( V SPaths  E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) )
1917, 18syl 16 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  -> 
( F ( V SPaths  E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) )
206, 9, 19mpbir2and 913 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  F ( V SPaths  E
) P )
2120ex 434 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( I  =/= 
J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F ( V SPaths  E
) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   {cpr 3988   {ctp 3990   <.cop 3992   class class class wbr 4401   `'ccnv 4948   Fun wfun 5521   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395   2c2 10483   Trails ctrail 23559   SPaths cspath 23561
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-wlk 23568  df-trail 23569  df-spth 23571
This theorem is referenced by:  usgra2adedgspth  30454
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