Theorem constr2wlk 23465

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 23419	. . . . . . . 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 12231	. . . . . . 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 23425	. . . . 5 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... 2 ) --> V )
13	2, 3	2trllemA 23417	. . . . . . 7 |- ( # ` F ) = 2
14	13	oveq2i 6097	. . . . . 6 |- ( 0 ... ( # ` F ) ) = ( 0 ... 2 )
15	14	feq2i 5547	. . . . 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 23464	. . . 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 23418	. . . . . 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 2918	. . . 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 4529	. . . . . 6 |- { <. 0 , I >. , <. 1 , J >. } e. _V
24	3, 23	eqeltri 2508	. . . . 5 |- F e. _V
25		tpex 6374	. . . . . 6 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
26	11, 25	eqeltri 2508	. . . . 5 |- P e. _V
27		iswlk 23394	. . . . 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 1369   e. wcel 1756   A.wral 2710   _Vcvv 2967   {cpr 3874   {ctp 3876   <.cop 3878   class class class wbr 4287   dom cdm 4835   -->wf 5409   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275   + caddc 9277   2c2 10363   ...cfz 11429  ..^cfzo 11540   #chash 12095  Word cword 12213   Walks cwalk 23373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-wlk 23383

This theorem is referenced by:  usgra2adedgwlk  30277  usgra2adedgwlkon  30278