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.

Step	Hyp	Ref	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 >. }
4	2, 3	2trllemH 24675	. . . . . . . 8 |- ( ( ( V e. X /\ E e. Y /\ B e. V ) /\ ( ( E ` I ) = { A , B } /\ ( E ` J ) = { B , C } ) ) -> F : ( 0..^ ( # ` F ) ) --> dom E )
5	1, 4	sylanbr 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 )
7	5, 6	syl 16	. . . . . 6 |- ( ( ( ( V e. X /\ E e. Y ) /\ B e. V ) /\ ( ( E ` I ) = { A , B } /\ ( E ` J ) = { B , C } ) ) -> F e. Word dom E )
8	7	ex 432	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ B e. V ) -> ( ( ( E ` I ) = { A , B } /\ ( E ` J ) = { B , C } ) -> F e. Word dom E ) )
9	8	3ad2antr2 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 ) )
10	9	imp 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 >. }
12	11	2trllemG 24681	. . . . 5 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... 2 ) --> V )
13	2, 3	2trllemA 24673	. . . . . . 7 |- ( # ` F ) = 2
14	13	oveq2i 6207	. . . . . 6 |- ( 0 ... ( # ` F ) ) = ( 0 ... 2 )
15	14	feq2i 5632	. . . . 5 |- ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V )
16	12, 15	sylibr 212	. . . 4 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... ( # ` F ) ) --> V )
17	16	ad2antlr 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 )
18	2, 3, 11	2wlklem1 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 ) ) } )
19	2, 3	2trllemB 24674	. . . . . 6 |- ( 0..^ ( # ` F ) ) = { 0 , 1 }
20	19	a1i 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 } )
21	20	raleqdv 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 ) ) } ) )
22	18, 21	mpbird 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 ) ) } ) ) )
24	23	ad2antrr 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 ) ) } ) ) )
25	10, 17, 22, 24	mpbir3and 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 )
26	25	ex 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 ) )

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