Theorem clwwlknon 16875

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 16874	. . . 4 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
2	1	elmpocl 6287	. . 3 |- ( x e. ( X (ClWWalksNOn ` G ) N ) -> ( X e. (Vtx ` G ) /\ N e. NN0 ) )
3		fveq1 5697	. . . . . . . 8 |- ( w = x -> ( w ` 0 ) = ( x ` 0 ) )
4	3	eqeq1d 2247	. . . . . . 7 |- ( w = x -> ( ( w ` 0 ) = X <-> ( x ` 0 ) = X ) )
5	4	elrab 2982	. . . . . 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 2979	. . . . . . . 8 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x e. ( N ClWWalksN G ) )
8		clwwlkclwwlkn 16855	. . . . . . . 8 |- ( x e. ( N ClWWalksN G ) -> x e. (ClWWalks ` G ) )
9		eqid 2238	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
10	9	clwwlkbp 16841	. . . . . . . 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 1041	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x e. Word (Vtx ` G ) )
13	11	simp3d 1042	. . . . . 6 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> x =/= (/) )
14		fstwrdne 11365	. . . . . 6 |- ( ( x e. Word (Vtx ` G ) /\ x =/= (/) ) -> ( x ` 0 ) e. (Vtx ` G ) )
15	12, 13, 14	syl2anc 415	. . . . 5 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> ( x ` 0 ) e. (Vtx ` G ) )
16	6, 15	eqeltrrd 2316	. . . 4 |- ( x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } -> X e. (Vtx ` G ) )
17		clwwlknnn 16858	. . . . . 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 9631	. . . 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 16844	. . . . . . . . . 10 |- ( g e. _V -> (ClWWalks ` g ) e. _V )
22	21	elv 2825	. . . . . . . . 9 |- (ClWWalks ` g ) e. _V
23	22	rabex 4282	. . . . . . . 8 |- { w e. (ClWWalks ` g ) | (♯ ` w ) = n } e. _V
24	23	gen2 1503	. . . . . . 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 16466	. . . . . . . 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 16850	. . . . . . . 8 |- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. (ClWWalks ` g ) | (♯ ` w ) = n } )
29	28	mpofvex 6445	. . . . . . 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 1383	. . . . . 6 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( N ClWWalksN G ) e. _V )
31		rabexg 4281	. . . . . 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 2248	. . . . . . 7 |- ( v = X -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = X ) )
34	33	rabbidv 2810	. . . . . 6 |- ( v = X -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } )
35		oveq1 6095	. . . . . . 7 |- ( n = N -> ( n ClWWalksN G ) = ( N ClWWalksN G ) )
36	35	rabeqdv 2815	. . . . . 6 |- ( n = N -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
37	34, 36, 1	ovmpog 6226	. . . . 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 1379	. . . 4 |- ( ( X e. (Vtx ` G ) /\ N e. NN0 ) -> ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
39	38	eleq2d 2308	. . 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 716	. 2 |- ( x e. ( X (ClWWalksNOn ` G ) N ) <-> x e. { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X } )
41	40	eqriv 2235	1 |- ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X }

This proof depends on syntax axioms:   /\ wa 104   /\ w3a 1009   A.wal 1400   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   (/)c0 3520   ` cfv 5380  (class class class)co 6088   0cc0 8185   NNcn 9313   NN0cn0 9574  ♯chash 11236  Word cword 11326  Vtxcvtx 16458  ClWWalkscclwwlk 16837   ClWWalksN cclwwlkn 16849  ClWWalksNOncclwwlknon 16872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-map 6928  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-ndx 13413  df-slot 13414  df-base 13416  df-vtx 16460  df-clwwlk 16838  df-clwwlkn 16850  df-clwwlknon 16873

This theorem is used by:  isclwwlknon  16876  clwwlknon2  16880