Theorem clwwlksnun 26974

Description: The set of closed walks of fixed length in a simple graph is the union of the closed walks of the fixed length starting at each of the vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.)

Hypothesis

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

Assertion

Ref	Expression
clwwlksnun	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥})

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

Allowed substitution hint:   𝑉(𝑤)

Proof of Theorem clwwlksnun

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

Step	Hyp	Ref	Expression
1		eliun 4524	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∃𝑥 ∈ 𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥})
2		fveq1 6190	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
3	2	eqeq1d 2624	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑦‘0) = 𝑥))
4	3	elrab 3363	. . . . 5 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
5	4	rexbii 3041	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
6		simpl 473	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
7	6	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
8	7	rexlimdvw 3034	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
9		clwwlksnun.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
10		eqid 2622	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
11	9, 10	clwwlknp 26887	. . . . . . . 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)))
12	11	anim2i 593	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))))
13	10, 9	usgrpredgv 26089	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → (( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉))
14	13	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉)))
15		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉) → (𝑦‘0) ∈ 𝑉)
16	14, 15	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝑦‘0) ∈ 𝑉))
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝑦‘0) ∈ 𝑉))
18	17	com12 32	. . . . . . . . . . 11 ⊢ ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦‘0) ∈ 𝑉))
19	18	3ad2ant3 1084	. . . . . . . . . 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 ∧ 𝑁 ∈ ℕ0) → (𝑦‘0) ∈ 𝑉))
20	19	impcom 446	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((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)
21		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((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))
22	21	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((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)
23	22	biantrud 528	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((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)))
24	23	bicomd 213	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((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)))
25	20, 24	rspcedv 3313	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((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)))
26	25	adantld 483	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
27	12, 26	mpcom 38	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
28	27	ex 450	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
29	8, 28	impbid 202	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
30	5, 29	syl5bb 272	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥 ∈ 𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
31	1, 30	syl5rbb 273	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}))
32	31	eqrdv 2620	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥})

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  {cpr 4179  ∪ ciun 4520  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939   − cmin 10266  ℕ₀cn0 11292  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-umgr 25978  df-usgr 26046  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlk4  27244