Theorem clwwlknun 16562

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 4000	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁))
2		isclwwlknon 16551	. . . . 5 ⊢ (𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
3	2	rexbii 2551	. . . 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 2658	. . . . 5 ⊢ (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
6		clwwlknun.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2234	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
8	6, 7	clwwlknp 16538	. . . . . . . 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 16319	. . . . . . . . . . . . 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 1047	. . . . . . . . . 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 2240	. . . . . . . . . . 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 2927	. . . . . . . 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 2232	1 ⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {cpr 3695  ∪ ciun 3996  ‘cfv 5357  (class class class)co 6058  0cc0 8143  1c1 8144   + caddc 8146   − cmin 8460  ..^cfzo 10498  ♯chash 11163  Word cword 11249  lastSclsw 11294  Vtxcvtx 16133  Edgcedg 16178  USGraphcusgr 16275   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-2o 6661  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-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  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-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-umgren 16215  df-usgren 16277  df-clwwlk 16513  df-clwwlkn 16525  df-clwwlknon 16548

This theorem is referenced by: (None)