Theorem wwlknprop 24362

Description: Properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.)

Assertion

Ref	Expression
wwlknprop	|- ( P e. ( ( V WWalksN E ) ` N ) -> ( ( V e. _V /\ E e. _V ) /\ ( N e. NN0 /\ P e. Word V ) ) )

Proof of Theorem wwlknprop

Dummy variables e n v w i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wwlkn 24356	. . 3 |- WWalksN = ( v e. _V , e e. _V |-> ( n e. NN0 |-> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } ) )
2		df-rab 2823	. . . . . . 7 |- { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } = { w | ( w e. ( v WWalks e ) /\ ( # ` w ) = ( n + 1 ) ) }
3		iswwlk 24359	. . . . . . . . . 10 |- ( ( v e. _V /\ e e. _V ) -> ( w e. ( v WWalks e ) <-> ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) ) )
4	3	adantr 465	. . . . . . . . 9 |- ( ( ( v e. _V /\ e e. _V ) /\ n e. NN0 ) -> ( w e. ( v WWalks e ) <-> ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) ) )
5	4	anbi1d 704	. . . . . . . 8 |- ( ( ( v e. _V /\ e e. _V ) /\ n e. NN0 ) -> ( ( w e. ( v WWalks e ) /\ ( # ` w ) = ( n + 1 ) ) <-> ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) ) )
6	5	abbidv 2603	. . . . . . 7 |- ( ( ( v e. _V /\ e e. _V ) /\ n e. NN0 ) -> { w | ( w e. ( v WWalks e ) /\ ( # ` w ) = ( n + 1 ) ) } = { w | ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } )
7	2, 6	syl5eq 2520	. . . . . 6 |- ( ( ( v e. _V /\ e e. _V ) /\ n e. NN0 ) -> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } = { w | ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } )
8		3anan12 986	. . . . . . . . . 10 |- ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) <-> ( w e. Word v /\ ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) ) )
9	8	anbi1i 695	. . . . . . . . 9 |- ( ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) <-> ( ( w e. Word v /\ ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) ) /\ ( # ` w ) = ( n + 1 ) ) )
10		anass 649	. . . . . . . . 9 |- ( ( ( w e. Word v /\ ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) ) /\ ( # ` w ) = ( n + 1 ) ) <-> ( w e. Word v /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) ) )
11	9, 10	bitri 249	. . . . . . . 8 |- ( ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) <-> ( w e. Word v /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) ) )
12	11	abbii 2601	. . . . . . 7 |- { w | ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } = { w | ( w e. Word v /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) ) }
13		df-rab 2823	. . . . . . 7 |- { w e. Word v | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } = { w | ( w e. Word v /\ ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) ) }
14	12, 13	eqtr4i 2499	. . . . . 6 |- { w | ( ( w =/= (/) /\ w e. Word v /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } = { w e. Word v | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) }
15	7, 14	syl6eq 2524	. . . . 5 |- ( ( ( v e. _V /\ e e. _V ) /\ n e. NN0 ) -> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } = { w e. Word v | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } )
16	15	mpteq2dva 4533	. . . 4 |- ( ( v e. _V /\ e e. _V ) -> ( n e. NN0 |-> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } ) = ( n e. NN0 |-> { w e. Word v | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } ) )
17	16	mpt2eq3ia 6344	. . 3 |- ( v e. _V , e e. _V |-> ( n e. NN0 |-> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } ) ) = ( v e. _V , e e. _V |-> ( n e. NN0 |-> { w e. Word v | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } ) )
18	1, 17	eqtri 2496	. 2 |- WWalksN = ( v e. _V , e e. _V |-> ( n e. NN0 |-> { w e. Word v | ( ( w =/= (/) /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e ) /\ ( # ` w ) = ( n + 1 ) ) } ) )
19	18	elovmptnn0wrd 12545	1 |- ( P e. ( ( V WWalksN E ) ` N ) -> ( ( V e. _V /\ E e. _V ) /\ ( N e. NN0 /\ P e. Word V ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   (/)c0 3785   {cpr 4029   |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   0cc0 9488   1c1 9489   + caddc 9491   - cmin 9801   NN0cn0 10791  ..^cfzo 11788   #chash 12369  Word cword 12496   WWalks cwwlk 24353   WWalksN cwwlkn 24354

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-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-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-word 12504  df-wwlk 24355  df-wwlkn 24356

This theorem is referenced by:  wwlknimp  24363  wwlkn0  24365  wlklniswwlkn2  24376  wwlkiswwlkn  24378  wwlknred  24399  wwlknext  24400  wwlkextwrd  24404  wwlkextsur  24407  wwlkextbij0  24408  wwlkexthasheq  24410  wwlknndef  24413  wwlkextproplem3  24419  wwlkext2clwwlk  24479  numclwwlk2lem1  24779