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

Theorem clwwlknonclwlknonen 27548
 Description: The sets of the two representations of closed walks of a fixed positive length on a fixed vertex are equinumerous. (Contributed by AV, 27-May-2022.)
Assertion
Ref Expression
clwwlknonclwlknonen ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑋

Proof of Theorem clwwlknonclwlknonen
Dummy variables 𝑐 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6342 . . 3 (ClWalks‘𝐺) ∈ V
21rabex 4943 . 2 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ V
3 ovex 6822 . 2 (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
4 eqid 2770 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
5 fveq2 6332 . . . . . . 7 (𝑤 = 𝑢 → (1st𝑤) = (1st𝑢))
65fveq2d 6336 . . . . . 6 (𝑤 = 𝑢 → (♯‘(1st𝑤)) = (♯‘(1st𝑢)))
76eqeq1d 2772 . . . . 5 (𝑤 = 𝑢 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑢)) = 𝑁))
8 fveq2 6332 . . . . . . 7 (𝑤 = 𝑢 → (2nd𝑤) = (2nd𝑢))
98fveq1d 6334 . . . . . 6 (𝑤 = 𝑢 → ((2nd𝑤)‘0) = ((2nd𝑢)‘0))
109eqeq1d 2772 . . . . 5 (𝑤 = 𝑢 → (((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑢)‘0) = 𝑋))
117, 10anbi12d 608 . . . 4 (𝑤 = 𝑢 → (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑋)))
1211cbvrabv 3348 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑢 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑢)) = 𝑁 ∧ ((2nd𝑢)‘0) = 𝑋)}
13 eqid 2770 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩))
144, 12, 13clwwlknonclwlknonf1o 27547 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
15 f1oen2g 8125 . 2 (({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ V ∧ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V ∧ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))
162, 3, 14, 15mp3an12i 1575 1 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  {crab 3064  Vcvv 3349  ⟨cop 4320   class class class wbr 4784   ↦ cmpt 4861  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313   ≈ cen 8105  0cc0 10137  ℕcn 11221  ♯chash 13320   substr csubstr 13490  Vtxcvtx 26094  USPGraphcuspgr 26264  ClWalkscclwlks 26900  ClWWalksNOncclwwlknon 27256 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ifp 1049  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-rp 12035  df-fz 12533  df-fzo 12673  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-uspgr 26266  df-wlks 26729  df-clwlks 26901  df-clwwlk 27129  df-clwwlkn 27173  df-clwwlknon 27257 This theorem is referenced by:  clwlknon2num  27554  numclwlk1lem2  27556
 Copyright terms: Public domain W3C validator