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Theorem wlksnwwlknvbij 26706
Description: There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 20-Apr-2021.)
Assertion
Ref Expression
wlksnwwlknvbij ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
Distinct variable groups:   𝑓,𝐺,𝑝,𝑤   𝑓,𝑁,𝑝,𝑤   𝑓,𝑋,𝑝,𝑤

Proof of Theorem wlksnwwlknvbij
Dummy variables 𝑞 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6168 . . . . 5 (Walks‘𝐺) ∈ V
21mptrabex 6453 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ∈ V
32resex 5412 . . 3 ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V
4 eqid 2621 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝))
5 usgruspgr 26000 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
6 fveq2 6158 . . . . . . . . . 10 (𝑞 = 𝑡 → (1st𝑞) = (1st𝑡))
76fveq2d 6162 . . . . . . . . 9 (𝑞 = 𝑡 → (#‘(1st𝑞)) = (#‘(1st𝑡)))
87eqeq1d 2623 . . . . . . . 8 (𝑞 = 𝑡 → ((#‘(1st𝑞)) = 𝑁 ↔ (#‘(1st𝑡)) = 𝑁))
98cbvrabv 3189 . . . . . . 7 {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} = {𝑡 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑡)) = 𝑁}
10 eqid 2621 . . . . . . 7 (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
11 fveq2 6158 . . . . . . . 8 (𝑝 = 𝑠 → (2nd𝑝) = (2nd𝑠))
1211cbvmptv 4720 . . . . . . 7 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑠 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑠))
139, 10, 12wlknwwlksnbij 26680 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
145, 13sylan 488 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
15143adant3 1079 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
16 fveq1 6157 . . . . . 6 (𝑤 = (2nd𝑝) → (𝑤‘0) = ((2nd𝑝)‘0))
1716eqeq1d 2623 . . . . 5 (𝑤 = (2nd𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
18173ad2ant3 1082 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) ∧ 𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ 𝑤 = (2nd𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
194, 15, 18f1oresrab 6361 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
20 f1oeq1 6094 . . . 4 (𝑓 = ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) → (𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
2120spcegv 3284 . . 3 (((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V → (((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
223, 19, 21mpsyl 68 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
23 df-rab 2917 . . . . 5 {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋))}
24 anass 680 . . . . . . 7 (((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)))
2524bicomi 214 . . . . . 6 ((𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)) ↔ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋))
2625abbii 2736 . . . . 5 {𝑝 ∣ (𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋))} = {𝑝 ∣ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋)}
27 fveq2 6158 . . . . . . . . . . . 12 (𝑞 = 𝑝 → (1st𝑞) = (1st𝑝))
2827fveq2d 6162 . . . . . . . . . . 11 (𝑞 = 𝑝 → (#‘(1st𝑞)) = (#‘(1st𝑝)))
2928eqeq1d 2623 . . . . . . . . . 10 (𝑞 = 𝑝 → ((#‘(1st𝑞)) = 𝑁 ↔ (#‘(1st𝑝)) = 𝑁))
3029elrab 3351 . . . . . . . . 9 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↔ (𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁))
3130anbi1i 730 . . . . . . . 8 ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋) ↔ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋))
3231bicomi 214 . . . . . . 7 (((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋))
3332abbii 2736 . . . . . 6 {𝑝 ∣ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋)}
34 df-rab 2917 . . . . . 6 {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋)}
3533, 34eqtr4i 2646 . . . . 5 {𝑝 ∣ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
3623, 26, 353eqtri 2647 . . . 4 {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
37 f1oeq2 6095 . . . 4 ({𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
3836, 37mp1i 13 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
3938exbidv 1847 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → (∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
4022, 39mpbird 247 1 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  {crab 2912  Vcvv 3190  cmpt 4683  cres 5086  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  0cc0 9896  0cn0 11252  #chash 13073  Vtxcvtx 25808   USPGraph cuspgr 25970   USGraph cusgr 25971  Walkscwlks 26396   WWalksN cwwlksn 26621
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-edg 25874  df-uhgr 25883  df-upgr 25907  df-uspgr 25972  df-usgr 25973  df-wlks 26399  df-wwlks 26625  df-wwlksn 26626
This theorem is referenced by:  rusgrnumwlkg  26773
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