Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwwlksnun Structured version   Visualization version   GIF version

Theorem clwwlksnun 41281
 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 4460 . . 3 (𝑦 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∃𝑥𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥})
2 fveq1 6102 . . . . . . 7 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
32eqeq1d 2612 . . . . . 6 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑦‘0) = 𝑥))
43elrab 3331 . . . . 5 (𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑦‘0) = 𝑥))
54rexbii 3023 . . . 4 (∃𝑥𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑦‘0) = 𝑥))
6 simpl 472 . . . . . . 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 3016 . . . . 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 2610 . . . . . . . . 9 (Edg‘𝐺) = (Edg‘𝐺)
119, 10clwwlknp 41195 . . . . . . . 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 591 . . . . . . 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, 9usgrpredgav 40424 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → (( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉))
1413ex 449 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉)))
15 simpr 476 . . . . . . . . . . . . . 14 ((( lastS ‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉) → (𝑦‘0) ∈ 𝑉)
1614, 15syl6 34 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → ({( lastS ‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝑦‘0) ∈ 𝑉))
1716adantr 480 . . . . . . . . . . . 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 1077 . . . . . . . . . 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 445 . . . . . . . . 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 476 . . . . . . . . . . . 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 2616 . . . . . . . . . . 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 527 . . . . . . . . . 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 212 . . . . . . . . 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 3286 . . . . . . . 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 482 . . . . . . 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 37 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ (𝑁 ClWWalkSN 𝐺)) → ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑦‘0) = 𝑥))
2827ex 449 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) → ∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑦‘0) = 𝑥)))
298, 28impbid 201 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥𝑉 (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalkSN 𝐺)))
305, 29syl5bb 271 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑥𝑉 𝑦 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ 𝑦 ∈ (𝑁 ClWWalkSN 𝐺)))
311, 30syl5rbb 272 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) ↔ 𝑦 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥}))
3231eqrdv 2608 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  {cpr 4127  ∪ ciun 4455  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   ClWWalkSN cclwwlksn 41184 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-umgr 25750  df-edga 25793  df-usgr 40381  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  av-numclwwlk4  41540
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