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Theorem constr2trl 24263
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 1171 . . . . . . . 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 997 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  I  =/=  J )
8 3simpc 990 . . . . . . . 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 24217 . . . . . . 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 1223 . . . . . 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 1157 . . . 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 24222 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 2 ) --> V )
1910, 112trllemA 24214 . . . . . . . 8  |-  ( # `  F )  =  2
2019oveq2i 6286 . . . . . . 7  |-  ( 0 ... ( # `  F
) )  =  ( 0 ... 2 )
2120feq2i 5715 . . . . . 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 24261 . . . . 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 24215 . . . . . 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 3057 . . . 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 4682 . . . . . . 7  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
3211, 31eqeltri 2544 . . . . . 6  |-  F  e. 
_V
33 tpex 6574 . . . . . . 7  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
3417, 33eqeltri 2544 . . . . . 6  |-  P  e. 
_V
35 istrl2 24202 . . . . . 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 1174 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   _Vcvv 3106   {cpr 4022   {ctp 4024   <.cop 4026   class class class wbr 4440   dom cdm 4992   -->wf 5575   -1-1->wf1 5576   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   2c2 10574   ...cfz 11661  ..^cfzo 11781   #chash 12360   Trails ctrail 24161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-wlk 24170  df-trail 24171
This theorem is referenced by:  constr2spth  24264  constr2pth  24265  2pthon  24266
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