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Theorem clwwlkf1o 27846
 Description: F is a 1-1 onto function, that means that there is a bijection between the set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwwlkf1o.d 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
clwwlkf1o.f 𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))
Assertion
Ref Expression
clwwlkf1o (𝑁 ∈ ℕ → 𝐹:𝐷1-1-onto→(𝑁 ClWWalksN 𝐺))
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑡,𝐷   𝑡,𝐺,𝑤   𝑡,𝑁
Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)

Proof of Theorem clwwlkf1o
StepHypRef Expression
1 clwwlkf1o.d . . 3 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
2 clwwlkf1o.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))
31, 2clwwlkf1 27844 . 2 (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺))
41, 2clwwlkfo 27845 . 2 (𝑁 ∈ ℕ → 𝐹:𝐷onto→(𝑁 ClWWalksN 𝐺))
5 df-f1o 6332 . 2 (𝐹:𝐷1-1-onto→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺) ∧ 𝐹:𝐷onto→(𝑁 ClWWalksN 𝐺)))
63, 4, 5sylanbrc 586 1 (𝑁 ∈ ℕ → 𝐹:𝐷1-1-onto→(𝑁 ClWWalksN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110   ↦ cmpt 5111  –1-1→wf1 6322  –onto→wfo 6323  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136  0cc0 10529  ℕcn 11628  lastSclsw 13908   prefix cpfx 14026   WWalksN cwwlksn 27622   ClWWalksN cclwwlkn 27819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-xnn0 11959  df-z 11973  df-uz 12235  df-rp 12381  df-fz 12889  df-fzo 13032  df-hash 13690  df-word 13861  df-lsw 13909  df-concat 13917  df-s1 13944  df-substr 13997  df-pfx 14027  df-wwlks 27626  df-wwlksn 27627  df-clwwlk 27777  df-clwwlkn 27820 This theorem is referenced by:  clwwlken  27847  clwwlkvbij  27908
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