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.

Step	Hyp	Ref	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 >. }
4	2, 3	2trllemH 24227	. . . . . . . 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 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 )
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 434	. . . . 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 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 ) )
10	9	imp 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 >. }
12	11	2trllemG 24233	. . . . 5 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... 2 ) --> V )
13	2, 3	2trllemA 24225	. . . . . . 7 |- ( # ` F ) = 2
14	13	oveq2i 6293	. . . . . 6 |- ( 0 ... ( # ` F ) ) = ( 0 ... 2 )
15	14	feq2i 5722	. . . . 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 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 )
18	2, 3, 11	2wlklem1 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 ) ) } )
19	2, 3	2trllemB 24226	. . . . . 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 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 ) ) } ) )
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		prex 4689	. . . . . 6 |- { <. 0 , I >. , <. 1 , J >. } e. _V
24	3, 23	eqeltri 2551	. . . . 5 |- F e. _V
25		tpex 6581	. . . . . 6 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
26	11, 25	eqeltri 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 ) ) } ) ) )
28	24, 26, 27	mpanr12 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 ) ) } ) ) )
29	28	ad2antrr 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 ) ) } ) ) )
30	10, 17, 22, 29	mpbir3and 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 )
31	30	ex 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 ) )

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