Theorem constr2wlk 25407

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 1009	. . . . . . . 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 25361	. . . . . . . 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 481	. . . . . . 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 12722	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) --> dom E -> F e. Word dom E )
7	5, 6	syl 17	. . . . . 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 441	. . . . 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 1196	. . . 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 436	. . 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 25367	. . . . 5 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... 2 ) --> V )
13	2, 3	2trllemA 25359	. . . . . . 7 |- ( # ` F ) = 2
14	13	oveq2i 6319	. . . . . 6 |- ( 0 ... ( # ` F ) ) = ( 0 ... 2 )
15	14	feq2i 5731	. . . . 5 |- ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V )
16	12, 15	sylibr 217	. . . 4 |- ( ( A e. V /\ B e. V /\ C e. V ) -> P : ( 0 ... ( # ` F ) ) --> V )
17	16	ad2antlr 741	. . 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 25406	. . . 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 25360	. . . . . 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 2979	. . . 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 240	. . 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 25331	. . . 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 740	. . 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 1213	. 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 441	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   {cpr 3961   {ctp 3963   <.cop 3965   class class class wbr 4395   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   + caddc 9560   2c2 10681   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703   Walks cwalk 25305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-wlk 25315

This theorem is referenced by:  usgra2adedgwlk  25421  usgra2adedgwlkon  25422