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Theorem constr2trl 23496
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 1168 . . . . . . . 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 994 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  I  =/=  J )
8 3simpc 987 . . . . . . . 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 23450 . . . . . . 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 1218 . . . . . 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 1154 . . . 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 23455 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 2 ) --> V )
1910, 112trllemA 23447 . . . . . . . 8  |-  ( # `  F )  =  2
2019oveq2i 6100 . . . . . . 7  |-  ( 0 ... ( # `  F
) )  =  ( 0 ... 2 )
2120feq2i 5550 . . . . . 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 23494 . . . . 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 23448 . . . . . 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 2921 . . . 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 4532 . . . . . . 7  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
3211, 31eqeltri 2511 . . . . . 6  |-  F  e. 
_V
33 tpex 6377 . . . . . . 7  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
3417, 33eqeltri 2511 . . . . . 6  |-  P  e. 
_V
35 istrl2 23435 . . . . . 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 1171 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   _Vcvv 2970   {cpr 3877   {ctp 3879   <.cop 3881   class class class wbr 4290   dom cdm 4838   -->wf 5412   -1-1->wf1 5413   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281    + caddc 9283   2c2 10369   ...cfz 11435  ..^cfzo 11546   #chash 12101   Trails ctrail 23404
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-wlk 23413  df-trail 23414
This theorem is referenced by:  constr2spth  23497  constr2pth  23498  2pthon  23499
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