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Theorem constr2wlk 24721
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 973 . . . . . . . 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 24675 . . . . . . . 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 471 . . . . . . 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 12457 . . . . . . 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 432 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  B  e.  V )  ->  (
( ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F  e. Word  dom  E ) )
983ad2antr2 1160 . . . 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 427 . . 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 24681 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 2 ) --> V )
132, 32trllemA 24673 . . . . . . 7  |-  ( # `  F )  =  2
1413oveq2i 6207 . . . . . 6  |-  ( 0 ... ( # `  F
) )  =  ( 0 ... 2 )
1514feq2i 5632 . . . . 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 724 . . 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 24720 . . . 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 24674 . . . . . 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 2985 . . . 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 iswlkg 24645 . . . 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 ) ) } ) ) )
2423ad2antrr 723 . . 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 ) ) } ) ) )
2510, 17, 22, 24mpbir3and 1177 . 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 )
2625ex 432 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   {cpr 3946   {ctp 3948   <.cop 3950   class class class wbr 4367   dom cdm 4913   -->wf 5492   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406   2c2 10502   ...cfz 11593  ..^cfzo 11717   #chash 12307  Word cword 12438   Walks cwalk 24619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-wlk 24629
This theorem is referenced by:  usgra2adedgwlk  24735  usgra2adedgwlkon  24736
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