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Theorem clwwlksnun 26834
 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) = 𝑥𝑉 {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.
StepHypRef Expression
1 eliun 4495 . . 3 (𝑦 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∃𝑥𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥})
2 fveq1 6149 . . . . . . 7 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
32eqeq1d 2628 . . . . . 6 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑦‘0) = 𝑥))
43elrab 3351 . . . . 5 (𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
54rexbii 3039 . . . 4 (∃𝑥𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
6 simpl 473 . . . . . . 7 ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
76a1i 11 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
87rexlimdvw 3032 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
9 clwwlksnun.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
10 eqid 2626 . . . . . . . . 9 (Edg‘𝐺) = (Edg‘𝐺)
119, 10clwwlknp 26748 . . . . . . . 8 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)))
1211anim2i 592 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))))
1310, 9usgrpredgv 25976 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → (( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉))
1413ex 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)
1614, 15syl6 35 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝑦‘0) ∈ 𝑉))
1716adantr 481 . . . . . . . . . . . 12 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝑦‘0) ∈ 𝑉))
1817com12 32 . . . . . . . . . . 11 ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦‘0) ∈ 𝑉))
19183ad2ant3 1082 . . . . . . . . . 10 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦‘0) ∈ 𝑉))
2019impcom 446 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (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)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → 𝑥 = (𝑦‘0))
2221eqcomd 2632 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → (𝑦‘0) = 𝑥)
2322biantrud 528 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (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)))
2423bicomd 213 . . . . . . . . 9 ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (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)))
2520, 24rspcedv 3304 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
2625adantld 483 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
2712, 26mpcom 38 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
2827ex 450 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
298, 28impbid 202 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
305, 29syl5bb 272 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
311, 30syl5rbb 273 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑦 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}))
3231eqrdv 2624 1 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) = 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992  ∀wral 2912  ∃wrex 2913  {crab 2916  {cpr 4155  ∪ ciun 4490  ‘cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884   − cmin 10211  ℕ0cn0 11237  ..^cfzo 12403  #chash 13054  Word cword 13225   lastS clsw 13226  Vtxcvtx 25769  Edgcedg 25834   USGraph cusgr 25932   ClWWalksN cclwwlksn 26737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-edg 25835  df-umgr 25869  df-usgr 25934  df-clwwlks 26738  df-clwwlksn 26739 This theorem is referenced by:  numclwwlk4  27092
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