Theorem clwwlknon 16550

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 16549	. . . 4 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
2	1	elmpocl 6257	. . 3 |- ( x e. ( X (ClWWalksNOn ` G ) N ) -> ( X e. (Vtx ` G ) /\ N e. NN0 ) )
3		fveq1 5674	. . . . . . . 8 |- ( w = x -> ( w ` 0 ) = ( x ` 0 ) )
4	3	eqeq1d 2243	. . . . . . 7 |- ( w = x -> ( ( w ` 0 ) = X <-> ( x ` 0 ) = X ) )
5	4	elrab 2976	. . . . . 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 2973	. . . . . . . 8 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x e. ( N ClWWalksN G ) )
8		clwwlkclwwlkn 16530	. . . . . . . 8 |- ( x e. ( N ClWWalksN G ) -> x e. (ClWWalks ` G ) )
9		eqid 2234	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
10	9	clwwlkbp 16516	. . . . . . . 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 1037	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x e. Word (Vtx ` G ) )
13	11	simp3d 1038	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x =/= (/) )
14		fstwrdne 11288	. . . . . 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 2312	. . . 4 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> X e. (Vtx ` G ) )
17		clwwlknnn 16533	. . . . . 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 9570	. . . 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 16519	. . . . . . . . . 10 |- ( g e. _V -> (ClWWalks ` g ) e. _V )
22	21	elv 2819	. . . . . . . . 9 |- (ClWWalks ` g ) e. _V
23	22	rabex 4261	. . . . . . . 8 |- { w e. (ClWWalks ` g ) | (♯ ` w ) = n } e. _V
24	23	gen2 1499	. . . . . . 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 16141	. . . . . . . 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 16525	. . . . . . . 8 |- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. (ClWWalks ` g ) | (♯ ` w ) = n } )
29	28	mpofvex 6414	. . . . . . 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 1379	. . . . . 6 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( N ClWWalksN G ) e. _V )
31		rabexg 4260	. . . . . 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 2244	. . . . . . 7 |- ( v = X -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = X ) )
34	33	rabbidv 2804	. . . . . 6 |- ( v = X -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } )
35		oveq1 6065	. . . . . . 7 |- ( n = N -> ( n ClWWalksN G ) = ( N ClWWalksN G ) )
36	35	rabeqdv 2809	. . . . . 6 |- ( n = N -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
37	34, 36, 1	ovmpog 6196	. . . . 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 1375	. . . 4 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
39	38	eleq2d 2304	. . 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 712	. 2 |- ( x e. ( X (ClWWalksNOn ` G ) N ) <-> x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
41	40	eqriv 2231	1 |- ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X }

Syntax hints:   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   (/)c0 3512   ` cfv 5357  (class class class)co 6058   0cc0 8143   NNcn 9254   NN0cn0 9513  ♯chash 11163  Word cword 11249  Vtxcvtx 16133  ClWWalkscclwwlk 16512   ClWWalksN cclwwlkn 16524  ClWWalksNOncclwwlknon 16547

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlk 16513  df-clwwlkn 16525  df-clwwlknon 16548

This theorem is referenced by:  isclwwlknon  16551  clwwlknon2  16555