MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlksnwwlknvbij Structured version   Visualization version   GIF version

Theorem wlksnwwlknvbij 29839
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, 22-Jul-2018.) (Revised by AV, 5-Aug-2022.)
Assertion
Ref Expression
wlksnwwlknvbij ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃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 variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 fvex 6906 . . . . 5 (Walks‘𝐺) ∈ V
21mptrabex 7234 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ∈ V
32resex 6030 . . 3 ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V
4 eqid 2726 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝))
5 eqid 2726 . . . . 5 {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} = {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁}
6 eqid 2726 . . . . 5 (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
75, 6, 4wlknwwlksnbij 29819 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
8 fveq1 6892 . . . . . 6 (𝑤 = (2nd𝑝) → (𝑤‘0) = ((2nd𝑝)‘0))
98eqeq1d 2728 . . . . 5 (𝑤 = (2nd𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
1093ad2ant3 1132 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∧ 𝑤 = (2nd𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
114, 7, 10f1oresrab 7133 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((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})
12 f1oeq1 6823 . . . 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}))
1312spcegv 3582 . . 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}))
143, 11, 13mpsyl 68 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
15 2fveq3 6898 . . . . . . 7 (𝑝 = 𝑞 → (♯‘(1st𝑝)) = (♯‘(1st𝑞)))
1615eqeq1d 2728 . . . . . 6 (𝑝 = 𝑞 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑞)) = 𝑁))
1716rabrabi 3438 . . . . 5 {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}
1817eqcomi 2735 . . . 4 {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
19 f1oeq2 6824 . . . 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}))
2018, 19mp1i 13 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (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}))
2120exbidv 1917 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (∃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}))
2214, 21mpbird 256 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃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 205  wa 394   = wceq 1534  wex 1774  wcel 2099  {crab 3419  Vcvv 3462  cmpt 5228  cres 5676  1-1-ontowf1o 6545  cfv 6546  (class class class)co 7416  1st c1st 7993  2nd c2nd 7994  0cc0 11149  0cn0 12518  chash 14342  USPGraphcuspgr 29081  Walkscwlks 29530   WWalksN cwwlksn 29757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-oadd 8492  df-er 8726  df-map 8849  df-pm 8850  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-dju 9937  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-n0 12519  df-xnn0 12591  df-z 12605  df-uz 12869  df-fz 13533  df-fzo 13676  df-hash 14343  df-word 14518  df-edg 28981  df-uhgr 28991  df-upgr 29015  df-uspgr 29083  df-wlks 29533  df-wwlks 29761  df-wwlksn 29762
This theorem is referenced by:  rusgrnumwlkg  29908
  Copyright terms: Public domain W3C validator