Theorem clwwlknon 16214

Description: The set of closed walks on vertex X of length N in a graph G as words over the set of vertices. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)

Assertion

Ref	Expression
clwwlknon	|- ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X }

Distinct variable groups:   w, G   w, N   w, X

Proof of Theorem clwwlknon

Dummy variables n v x g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlknonmpo 16213	. . . 4 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
2	1	elmpocl 6210	. . 3 |- ( x e. ( X (ClWWalksNOn ` G ) N ) -> ( X e. (Vtx ` G ) /\ N e. NN0 ) )
3		fveq1 5632	. . . . . . . 8 |- ( w = x -> ( w ` 0 ) = ( x ` 0 ) )
4	3	eqeq1d 2238	. . . . . . 7 |- ( w = x -> ( ( w ` 0 ) = X <-> ( x ` 0 ) = X ) )
5	4	elrab 2960	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } <-> ( x e. ( N ClWWalksN G ) /\ ( x ` 0 ) = X ) )
6	5	simprbi 275	. . . . 5 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> ( x ` 0 ) = X )
7		elrabi 2957	. . . . . . . 8 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x e. ( N ClWWalksN G ) )
8		clwwlkclwwlkn 16194	. . . . . . . 8 |- ( x e. ( N ClWWalksN G ) -> x e. (ClWWalks ` G ) )
9		eqid 2229	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
10	9	clwwlkbp 16180	. . . . . . . 8 |- ( x e. (ClWWalks ` G ) -> ( G e. _V /\ x e. Word (Vtx ` G ) /\ x =/= (/) ) )
11	7, 8, 10	3syl 17	. . . . . . 7 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> ( G e. _V /\ x e. Word (Vtx ` G ) /\ x =/= (/) ) )
12	11	simp2d 1034	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x e. Word (Vtx ` G ) )
13	11	simp3d 1035	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x =/= (/) )
14		fstwrdne 11139	. . . . . 6 |- ( ( x e. Word (Vtx ` G ) /\ x =/= (/) ) -> ( x ` 0 ) e. (Vtx ` G ) )
15	12, 13, 14	syl2anc 411	. . . . 5 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> ( x ` 0 ) e. (Vtx ` G ) )
16	6, 15	eqeltrrd 2307	. . . 4 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> X e. (Vtx ` G ) )
17		clwwlknnn 16197	. . . . . 6 |- ( x e. ( N ClWWalksN G ) -> N e. NN )
18	7, 17	syl 14	. . . . 5 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> N e. NN )
19	18	nnnn0d 9443	. . . 4 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> N e. NN0 )
20	16, 19	jca 306	. . 3 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> ( X e. (Vtx ` G ) /\ N e. NN0 ) )
21		clwwlkex 16183	. . . . . . . . . 10 |- ( g e. _V -> (ClWWalks ` g ) e. _V )
22	21	elv 2804	. . . . . . . . 9 |- (ClWWalks ` g ) e. _V
23	22	rabex 4230	. . . . . . . 8 |- { w e. (ClWWalks ` g ) | (♯ ` w ) = n } e. _V
24	23	gen2 1496	. . . . . . 7 |- A. n A. g { w e. (ClWWalks ` g ) | (♯ ` w ) = n } e. _V
25		simpr 110	. . . . . . 7 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> N e. NN0 )
26	9	1vgrex 15858	. . . . . . . 8 |- ( X e. (Vtx ` G ) -> G e. _V )
27	26	adantr 276	. . . . . . 7 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> G e. _V )
28		df-clwwlkn 16189	. . . . . . . 8 |- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. (ClWWalks ` g ) | (♯ ` w ) = n } )
29	28	mpofvex 6363	. . . . . . 7 |- ( ( A. n A. g { w e. (ClWWalks ` g ) | (♯ ` w ) = n } e. _V /\ N e. NN0 /\ G e. _V ) -> ( N ClWWalksN G ) e. _V )
30	24, 25, 27, 29	mp3an2i 1376	. . . . . 6 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( N ClWWalksN G ) e. _V )
31		rabexg 4229	. . . . . 6 |- ( ( N ClWWalksN G ) e. _V -> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } e. _V )
32	30, 31	syl 14	. . . . 5 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } e. _V )
33		eqeq2 2239	. . . . . . 7 |- ( v = X -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = X ) )
34	33	rabbidv 2789	. . . . . 6 |- ( v = X -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } )
35		oveq1 6018	. . . . . . 7 |- ( n = N -> ( n ClWWalksN G ) = ( N ClWWalksN G ) )
36	35	rabeqdv 2794	. . . . . 6 |- ( n = N -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
37	34, 36, 1	ovmpog 6149	. . . . 5 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 /\ { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } e. _V ) -> ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
38	32, 37	mpd3an3 1372	. . . 4 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
39	38	eleq2d 2299	. . 3 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( x e. ( X (ClWWalksNOn ` G ) N ) <-> x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } ) )
40	2, 20, 39	pm5.21nii 709	. 2 |- ( x e. ( X (ClWWalksNOn ` G ) N ) <-> x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
41	40	eqriv 2226	1 |- ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X }

Syntax hints:   /\ wa 104   /\ w3a 1002   A.wal 1393   = wceq 1395   e. wcel 2200   =/= wne 2400   {crab 2512   _Vcvv 2800   (/)c0 3492   ` cfv 5322  (class class class)co 6011   0cc0 8020   NNcn 9131   NN0cn0 9390  ♯chash 11025  Word cword 11100  Vtxcvtx 15850  ClWWalkscclwwlk 16176   ClWWalksN cclwwlkn 16188  ClWWalksNOncclwwlknon 16211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-map 6812  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-ndx 13072  df-slot 13073  df-base 13075  df-vtx 15852  df-clwwlk 16177  df-clwwlkn 16189  df-clwwlknon 16212

This theorem is referenced by:  isclwwlknon  16215  clwwlknon2  16219