Theorem clwwlknun 16887

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 4016	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁))
2		isclwwlknon 16876	. . . . 5 ⊢ (𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
3	2	rexbii 2557	. . . 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 2664	. . . . 5 ⊢ (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
6		clwwlknun.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2238	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
8	6, 7	clwwlknp 16863	. . . . . . . 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 16644	. . . . . . . . . . . . 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 1051	. . . . . . . . . 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 2244	. . . . . . . . . . 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 2933	. . . . . . . 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 2236	1 ⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {cpr 3710  ∪ ciun 4012  ‘cfv 5380  (class class class)co 6088  0cc0 8185  1c1 8186   + caddc 8188   − cmin 8504  ..^cfzo 10566  ♯chash 11236  Word cword 11326  lastSclsw 11371  Vtxcvtx 16458  Edgcedg 16503  USGraphcusgr 16600   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-2o 6692  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-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  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-edgf 16451  df-vtx 16460  df-iedg 16461  df-edg 16504  df-umgren 16540  df-usgren 16602  df-clwwlk 16838  df-clwwlkn 16850  df-clwwlknon 16873

This theorem is used by: (None)