Theorem constr2wlk 23648

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 967	. . . . . . . 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 23602	. . . . . . . 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 12356	. . . . . . 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 1154	. . . 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 23608	. . . . 5 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... 2 ) --> V )
13	2, 3	2trllemA 23600	. . . . . . 7 |- ( # ` F ) = 2
14	13	oveq2i 6210	. . . . . 6 |- ( 0 ... ( # ` F ) ) = ( 0 ... 2 )
15	14	feq2i 5659	. . . . 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 23647	. . . 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 23601	. . . . . 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 3027	. . . 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 4641	. . . . . 6 |- { <. 0 , I >. , <. 1 , J >. } e. _V
24	3, 23	eqeltri 2538	. . . . 5 |- F e. _V
25		tpex 6488	. . . . . 6 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
26	11, 25	eqeltri 2538	. . . . 5 |- P e. _V
27		iswlk 23577	. . . . 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 1171	. 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 965   = wceq 1370   e. wcel 1758   A.wral 2798   _Vcvv 3076   {cpr 3986   {ctp 3988   <.cop 3990   class class class wbr 4399   dom cdm 4947   -->wf 5521   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393   + caddc 9395   2c2 10481   ...cfz 11553  ..^cfzo 11664   #chash 12219  Word cword 12338   Walks cwalk 23556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-wlk 23566

This theorem is referenced by:  usgra2adedgwlk  30453  usgra2adedgwlkon  30454