Theorem wwlknprop 30444

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 30438	. . 3 |- WWalksN = ( v e. _V , e e. _V |-> ( n e. NN0 |-> { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } ) )
2		df-rab 2801	. . . . . . 7 |- { w e. ( v WWalks e ) | ( # ` w ) = ( n + 1 ) } = { w | ( w e. ( v WWalks e ) /\ ( # ` w ) = ( n + 1 ) ) }
3		iswwlk 30441	. . . . . . . . . 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 2584	. . . . . . 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 2502	. . . . . 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 978	. . . . . . . . . 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 2582	. . . . . . 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 2801	. . . . . . 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 2481	. . . . . 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 2506	. . . . 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 4462	. . . 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 6236	. . 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 2478	. 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 30381	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 965   = wceq 1370   e. wcel 1757   {cab 2435   =/= wne 2641   A.wral 2792   {crab 2796   _Vcvv 3054   (/)c0 3721   {cpr 3963   |-> cmpt 4434   ran crn 4925   ` cfv 5502  (class class class)co 6176   |-> cmpt2 6178   0cc0 9369   1c1 9370   + caddc 9372   - cmin 9682   NN0cn0 10666  ..^cfzo 11635   #chash 12190  Word cword 12309   WWalks cwwlk 30435   WWalksN cwwlkn 30436

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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446

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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-word 12317  df-wwlk 30437  df-wwlkn 30438

This theorem is referenced by:  wwlknimp  30445  wwlkn0  30447  wlklniswwlkn2  30458  wwlkiswwlkn  30460  wwlknred  30479  wwlknext  30480  wwlkextwrd  30484  wwlkextsur  30487  wwlkextbij0  30488  wwlkexthasheq  30490  wwlknndef  30493  wwlkext2clwwlk  30589  wwlkextproplem3  30686  numclwwlk2lem1  30819