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

Theorem wlksnwwlknvbij 41114
 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 ∈ (1Walks‘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 6113 . . . . 5 (1Walks‘𝐺) ∈ V
21mptrabex 6392 . . . 4 (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ∈ V
32resex 5363 . . 3 ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V
4 eqid 2610 . . . 4 (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝))
5 usgruspgr 40408 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
6 fveq2 6103 . . . . . . . . . 10 (𝑞 = 𝑡 → (1st𝑞) = (1st𝑡))
76fveq2d 6107 . . . . . . . . 9 (𝑞 = 𝑡 → (#‘(1st𝑞)) = (#‘(1st𝑡)))
87eqeq1d 2612 . . . . . . . 8 (𝑞 = 𝑡 → ((#‘(1st𝑞)) = 𝑁 ↔ (#‘(1st𝑡)) = 𝑁))
98cbvrabv 3172 . . . . . . 7 {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} = {𝑡 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑡)) = 𝑁}
10 eqid 2610 . . . . . . 7 (𝑁 WWalkSN 𝐺) = (𝑁 WWalkSN 𝐺)
11 fveq2 6103 . . . . . . . 8 (𝑝 = 𝑠 → (2nd𝑝) = (2nd𝑠))
1211cbvmptv 4678 . . . . . . 7 (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑠 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑠))
139, 10, 12wlknwwlksnbij 41088 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))
145, 13sylan 487 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))
15143adant3 1074 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))
16 fveq1 6102 . . . . . 6 (𝑤 = (2nd𝑝) → (𝑤‘0) = ((2nd𝑝)‘0))
1716eqeq1d 2612 . . . . 5 (𝑤 = (2nd𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
18173ad2ant3 1077 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) ∧ 𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ 𝑤 = (2nd𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
194, 15, 18f1oresrab 6302 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
20 f1oeq1 6040 . . . 4 (𝑓 = ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) → (𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
2120spcegv 3267 . . 3 (((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V → (((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
223, 19, 21mpsyl 66 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
23 df-rab 2905 . . . . 5 {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋))}
24 anass 679 . . . . . . 7 (((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)))
2524bicomi 213 . . . . . 6 ((𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)) ↔ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋))
2625abbii 2726 . . . . 5 {𝑝 ∣ (𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋))} = {𝑝 ∣ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋)}
27 fveq2 6103 . . . . . . . . . . . 12 (𝑞 = 𝑝 → (1st𝑞) = (1st𝑝))
2827fveq2d 6107 . . . . . . . . . . 11 (𝑞 = 𝑝 → (#‘(1st𝑞)) = (#‘(1st𝑝)))
2928eqeq1d 2612 . . . . . . . . . 10 (𝑞 = 𝑝 → ((#‘(1st𝑞)) = 𝑁 ↔ (#‘(1st𝑝)) = 𝑁))
3029elrab 3331 . . . . . . . . 9 (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ↔ (𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁))
3130anbi1i 727 . . . . . . . 8 ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋) ↔ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋))
3231bicomi 213 . . . . . . 7 (((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋))
3332abbii 2726 . . . . . 6 {𝑝 ∣ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋)}
34 df-rab 2905 . . . . . 6 {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∧ ((2nd𝑝)‘0) = 𝑋)}
3533, 34eqtr4i 2635 . . . . 5 {𝑝 ∣ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑝)) = 𝑁) ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
3623, 26, 353eqtri 2636 . . . 4 {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
37 f1oeq2 6041 . . . 4 ({𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} → (𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘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 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
3938exbidv 1837 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → (∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
4022, 39mpbird 246 1 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ↦ cmpt 4643   ↾ cres 5040  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕ0cn0 11169  #chash 12979  Vtxcvtx 25673   USPGraph cuspgr 40378   USGraph cusgr 40379  1Walksc1wlks 40796   WWalkSN cwwlksn 41029 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-ifp 1007  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-2o 7448  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-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-1wlks 40800  df-wlks 40801  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  rusgrnumwlkg  41180
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