Theorem wwlknprop 24664

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 24658	. . 3 |- WWalksN = ( v e. _V , e e. _V |-> ( n e. NN0 |-> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } ) )
2		df-rab 2802	. . . . . . 7 |- { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } = { w | ( w e. ( v WWalks e ) /\ ( # ` w ) = ( n + 1 ) ) }
3		iswwlk 24661	. . . . . . . . . 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 2579	. . . . . . 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 2496	. . . . . 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 987	. . . . . . . . . 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 2577	. . . . . . 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 2802	. . . . . . 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 2475	. . . . . 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 2500	. . . . 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 4523	. . . 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 6347	. . 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 2472	. 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 12566	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 974   = wceq 1383   e. wcel 1804   {cab 2428   =/= wne 2638   A.wral 2793   {crab 2797   _Vcvv 3095   (/)c0 3770   {cpr 4016   |-> cmpt 4495   ran crn 4990   ` cfv 5578  (class class class)co 6281   |-> cmpt2 6283   0cc0 9495   1c1 9496   + caddc 9498   - cmin 9810   NN0cn0 10802  ..^cfzo 11806   #chash 12387  Word cword 12516   WWalks cwwlk 24655   WWalksN cwwlkn 24656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-word 12524  df-wwlk 24657  df-wwlkn 24658

This theorem is referenced by:  wwlknimp  24665  wwlkn0  24667  wlklniswwlkn2  24678  wwlkiswwlkn  24680  wwlknred  24701  wwlknext  24702  wwlkextwrd  24706  wwlkextsur  24709  wwlkextbij0  24710  wwlkexthasheq  24712  wwlknndef  24715  wwlkextproplem3  24721  wwlkext2clwwlk  24781  numclwwlk2lem1  25080