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Theorem constr2trl 24728
Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised 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
constr2trl  |-  ( ( ( 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 ) )

Proof of Theorem constr2trl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 simpll 753 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  V  e.  X )
2 simpr 461 . . . . . . . . . 10  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  Y )
32adantr 465 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  E  e.  Y )
4 simpr 461 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  B  e.  V )
51, 3, 43jca 1176 . . . . . . . 8  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  ( V  e.  X  /\  E  e.  Y  /\  B  e.  V )
)
65adantr 465 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  -> 
( V  e.  X  /\  E  e.  Y  /\  B  e.  V
) )
7 simpr1 1002 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  I  =/=  J )
8 3simpc 995 . . . . . . . 8  |-  ( ( I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  (
( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )
98adantl 466 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  -> 
( ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } ) )
10 2trlY.i . . . . . . . 8  |-  ( I  e.  U  /\  J  e.  W )
11 2trlY.f . . . . . . . 8  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
1210, 112trllemE 24682 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V
)  /\  I  =/=  J  /\  ( ( E `
 I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )
136, 7, 9, 12syl3anc 1228 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )
1413ex 434 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  (
( I  =/=  J  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  F : ( 0..^ (
# `  F )
) -1-1-> dom  E ) )
15143ad2antr2 1162 . . . 4  |-  ( ( ( 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 : ( 0..^ (
# `  F )
) -1-1-> dom  E ) )
1615imp 429 . . 3  |-  ( ( ( ( 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 : ( 0..^ (
# `  F )
) -1-1-> dom  E )
17 2trlY.p . . . . . . 7  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
18172trllemG 24687 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 2 ) --> V )
1910, 112trllemA 24679 . . . . . . . 8  |-  ( # `  F )  =  2
2019oveq2i 6307 . . . . . . 7  |-  ( 0 ... ( # `  F
) )  =  ( 0 ... 2 )
2120feq2i 5730 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V )
2218, 21sylibr 212 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... ( # `  F
) ) --> V )
2322adantl 466 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  ->  P : ( 0 ... ( # `  F
) ) --> V )
2423adantr 465 . . 3  |-  ( ( ( ( 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 }
) )  ->  P : ( 0 ... ( # `  F
) ) --> V )
2510, 11, 172wlklem1 24726 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
268, 25sylan2 474 . . . 4  |-  ( ( ( ( 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 }
) )  ->  A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
2710, 112trllemB 24680 . . . . . 6  |-  ( 0..^ ( # `  F
) )  =  {
0 ,  1 }
2827a1i 11 . . . . 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 }
) )  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
2928raleqdv 3060 . . . 4  |-  ( ( ( ( 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 }
) )  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  {
0 ,  1 }  ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
3026, 29mpbird 232 . . 3  |-  ( ( ( ( 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 }
) )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )
31 prex 4698 . . . . . . 7  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
3211, 31eqeltri 2541 . . . . . 6  |-  F  e. 
_V
33 tpex 6598 . . . . . . 7  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
3417, 33eqeltri 2541 . . . . . 6  |-  P  e. 
_V
35 istrl2 24667 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  _V  /\  P  e. 
_V ) )  -> 
( F ( V Trails  E ) P  <->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
3632, 34, 35mpanr12 685 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V Trails  E ) P  <->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
3736adantr 465 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( F ( V Trails  E ) P  <->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
3837adantr 465 . . 3  |-  ( ( ( ( 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  <->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
3916, 24, 30, 38mpbir3and 1179 . 2  |-  ( ( ( ( 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 )
4039ex 434 1  |-  ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456   dom cdm 5008   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   2c2 10606   ...cfz 11697  ..^cfzo 11821   #chash 12408   Trails ctrail 24626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-wlk 24635  df-trail 24636
This theorem is referenced by:  constr2spth  24729  constr2pth  24730  2pthon  24731
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