Theorem clwwlknun 16226

Description: The set of closed walks of fixed length 𝑁 in a simple graph 𝐺 is the union of the closed walks of the fixed length 𝑁 on each of the vertices of graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

Hypothesis

Ref	Expression
clwwlknun.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clwwlknun	⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Distinct variable groups:   𝑥,𝐺   𝑥,𝑁   𝑥,𝑉

Proof of Theorem clwwlknun

Dummy variables 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3970	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁))
2		isclwwlknon 16215	. . . . 5 ⊢ (𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
3	2	rexbii 2537	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
4		simpl 109	. . . . . 6 ⊢ ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
5	4	rexlimivw 2644	. . . . 5 ⊢ (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
6		clwwlknun.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2229	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
8	6, 7	clwwlknp 16202	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)))
9	8	anim2i 342	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → (𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))))
10	7, 6	usgrpredgv 16033	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → ((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉))
11	10	ex 115	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ({(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → ((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉)))
12		simpr 110	. . . . . . . . . . . 12 ⊢ (((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉) → (𝑦‘0) ∈ 𝑉)
13	11, 12	syl6com 35	. . . . . . . . . . 11 ⊢ ({(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝐺 ∈ USGraph → (𝑦‘0) ∈ 𝑉))
14	13	3ad2ant3 1044	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → (𝐺 ∈ USGraph → (𝑦‘0) ∈ 𝑉))
15	14	impcom 125	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (𝑦‘0) ∈ 𝑉)
16		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → 𝑥 = (𝑦‘0))
17	16	eqcomd 2235	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → (𝑦‘0) = 𝑥)
18	17	biantrud 304	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
19	18	bicomd 141	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
20	15, 19	rspcedv 2912	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
21	20	adantld 278	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
22	9, 21	mpcom 36	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
23	22	ex 115	. . . . 5 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
24	5, 23	impbid2 143	. . . 4 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
25	3, 24	bitrid 192	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
26	1, 25	bitr2id 193	. 2 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)))
27	26	eqrdv 2227	1 ⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {cpr 3668  ∪ ciun 3966  ‘cfv 5322  (class class class)co 6011  0cc0 8020  1c1 8021   + caddc 8023   − cmin 8338  ..^cfzo 10365  ♯chash 11025  Word cword 11100  lastSclsw 11145  Vtxcvtx 15850  Edgcedg 15895  USGraphcusgr 15989   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-2o 6576  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-2 9190  df-3 9191  df-4 9192  df-5 9193  df-6 9194  df-7 9195  df-8 9196  df-9 9197  df-n0 9391  df-z 9468  df-dec 9600  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-edgf 15843  df-vtx 15852  df-iedg 15853  df-edg 15896  df-umgren 15931  df-usgren 15991  df-clwwlk 16177  df-clwwlkn 16189  df-clwwlknon 16212

This theorem is referenced by: (None)