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Theorem wlksnwwlknvbij 28273
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 6787 . . . . 5 (Walks‘𝐺) ∈ V
21mptrabex 7101 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ∈ V
32resex 5939 . . 3 ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}) ∈ V
4 eqid 2738 . . . 4 (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝)) = (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ↦ (2nd𝑝))
5 eqid 2738 . . . . 5 {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} = {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁}
6 eqid 2738 . . . . 5 (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
75, 6, 4wlknwwlksnbij 28253 . . . 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 6773 . . . . . 6 (𝑤 = (2nd𝑝) → (𝑤‘0) = ((2nd𝑝)‘0))
98eqeq1d 2740 . . . . 5 (𝑤 = (2nd𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
1093ad2ant3 1134 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∧ 𝑤 = (2nd𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2nd𝑝)‘0) = 𝑋))
114, 7, 10f1oresrab 6999 . . 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 6704 . . . 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 3536 . . 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 6779 . . . . . . 7 (𝑝 = 𝑞 → (♯‘(1st𝑝)) = (♯‘(1st𝑞)))
1615eqeq1d 2740 . . . . . 6 (𝑝 = 𝑞 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑞)) = 𝑁))
1716rabrabi 3427 . . . . 5 {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋} = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}
1817eqcomi 2747 . . . 4 {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑞)) = 𝑁} ∣ ((2nd𝑝)‘0) = 𝑋}
19 f1oeq2 6705 . . . 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 1924 . 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 396   = wceq 1539  wex 1782  wcel 2106  {crab 3068  Vcvv 3432  cmpt 5157  cres 5591  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830  0cc0 10871  0cn0 12233  chash 14044  USPGraphcuspgr 27518  Walkscwlks 27963   WWalksN cwwlksn 28191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-edg 27418  df-uhgr 27428  df-upgr 27452  df-uspgr 27520  df-wlks 27966  df-wwlks 28195  df-wwlksn 28196
This theorem is referenced by:  rusgrnumwlkg  28342
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