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Theorem wlksnwwlknvbij 27281
 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 6459 . . . . 5 (Walks‘𝐺) ∈ V
21mptrabex 6760 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ∈ V
32resex 5693 . . 3 ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V
4 eqid 2778 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝))
5 eqid 2778 . . . . 5 {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} = {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁}
6 eqid 2778 . . . . 5 (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
75, 6, 4wlknwwlksnbij 27242 . . . 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 6445 . . . . . 6 (𝑤 = (2nd𝑝) → (𝑤‘0) = ((2nd𝑝)‘0))
98eqeq1d 2780 . . . . 5 (𝑤 = (2nd𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
1093ad2ant3 1126 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∧ 𝑤 = (2nd𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
114, 7, 10f1oresrab 6659 . . 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 6380 . . . 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 3496 . . 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 6451 . . . . . . 7 (𝑝 = 𝑞 → (♯‘(1st𝑝)) = (♯‘(1st𝑞)))
1615eqeq1d 2780 . . . . . 6 (𝑝 = 𝑞 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑞)) = 𝑁))
1716rabrabi 3397 . . . . 5 {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}
1817eqcomi 2787 . . . 4 {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
19 f1oeq2 6381 . . . 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 1964 . 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 249 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 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107  {crab 3094  Vcvv 3398   ↦ cmpt 4965   ↾ cres 5357  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  0cc0 10272  ℕ0cn0 11642  ♯chash 13435  USPGraphcuspgr 26497  Walkscwlks 26944   WWalksN cwwlksn 27175 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-edg 26396  df-uhgr 26406  df-upgr 26430  df-uspgr 26499  df-wlks 26947  df-wwlks 27179  df-wwlksn 27180 This theorem is referenced by:  rusgrnumwlkg  27358
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