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Theorem constr2wlk 24273
Description: Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-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
constr2wlk  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( ( E `
 I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
)  ->  F ( V Walks  E ) P ) )

Proof of Theorem constr2wlk
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 df-3an 975 . . . . . . . 8  |-  ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V )  <->  ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V ) )
2 2trlY.i . . . . . . . . 9  |-  ( I  e.  U  /\  J  e.  W )
3 2trlY.f . . . . . . . . 9  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
42, 32trllemH 24227 . . . . . . . 8  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
51, 4sylanbr 473 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
6 iswrdi 12512 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  F  e. Word  dom  E )
75, 6syl 16 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  B  e.  V )  /\  (
( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  F  e. Word  dom  E )
87ex 434 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  (
( ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F  e. Word  dom  E ) )
983ad2antr2 1162 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( ( E `
 I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
)  ->  F  e. Word  dom 
E ) )
109imp 429 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  F  e. Word  dom  E )
11 2trlY.p . . . . . 6  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
12112trllemG 24233 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 2 ) --> V )
132, 32trllemA 24225 . . . . . . 7  |-  ( # `  F )  =  2
1413oveq2i 6293 . . . . . 6  |-  ( 0 ... ( # `  F
) )  =  ( 0 ... 2 )
1514feq2i 5722 . . . . 5  |-  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V )
1612, 15sylibr 212 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... ( # `  F
) ) --> V )
1716ad2antlr 726 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  P : ( 0 ... ( # `  F
) ) --> V )
182, 3, 112wlklem1 24272 . . . 4  |-  ( ( ( ( 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 ) ) } )
192, 32trllemB 24226 . . . . . 6  |-  ( 0..^ ( # `  F
) )  =  {
0 ,  1 }
2019a1i 11 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
2120raleqdv 3064 . . . 4  |-  ( ( ( ( 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..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  {
0 ,  1 }  ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
2218, 21mpbird 232 . . 3  |-  ( ( ( ( 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..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )
23 prex 4689 . . . . . 6  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
243, 23eqeltri 2551 . . . . 5  |-  F  e. 
_V
25 tpex 6581 . . . . . 6  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
2611, 25eqeltri 2551 . . . . 5  |-  P  e. 
_V
27 iswlk 24193 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  _V  /\  P  e. 
_V ) )  -> 
( F ( V Walks 
E ) P  <->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2824, 26, 27mpanr12 685 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V Walks 
E ) P  <->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2928ad2antrr 725 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( F ( V Walks  E
) P  <->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
3010, 17, 22, 29mpbir3and 1179 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  F
( V Walks  E ) P )
3130ex 434 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( ( E `
 I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
)  ->  F ( V Walks  E ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   dom cdm 4999   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491   2c2 10581   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494   Walks cwalk 24171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-wlk 24181
This theorem is referenced by:  usgra2adedgwlk  24287  usgra2adedgwlkon  24288
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