MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rusgrnumwrdl2 Structured version   Visualization version   GIF version

Theorem rusgrnumwrdl2 27528
Description: In a k-regular simple graph, the number of edges resp. walks of length 1 (represented as words of length 2) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 6-May-2021.)
Hypothesis
Ref Expression
rusgrnumwrdl2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
rusgrnumwrdl2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
Distinct variable groups:   𝑤,𝐺   𝑤,𝑃   𝑤,𝑉
Allowed substitution hint:   𝐾(𝑤)

Proof of Theorem rusgrnumwrdl2
Dummy variables 𝑓 𝑝 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusgrnumwrdl2.v . . . . . 6 𝑉 = (Vtx‘𝐺)
21fvexi 6688 . . . . 5 𝑉 ∈ V
32wrdexi 13967 . . . 4 Word 𝑉 ∈ V
43rabex 5200 . . 3 {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))} ∈ V
52a1i 11 . . . 4 (𝐺 RegUSGraph 𝐾𝑉 ∈ V)
6 wrd2f1tovbij 14413 . . . 4 ((𝑉 ∈ V ∧ 𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}–1-1-onto→{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
75, 6sylan 583 . . 3 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}–1-1-onto→{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
8 hasheqf1oi 13804 . . 3 ({𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}–1-1-onto→{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)} → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})))
94, 7, 8mpsyl 68 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
101rusgrpropadjvtx 27527 . . . 4 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (♯‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
11 preq1 4624 . . . . . . . . 9 (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠})
1211eleq1d 2817 . . . . . . . 8 (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ (Edg‘𝐺) ↔ {𝑃, 𝑠} ∈ (Edg‘𝐺)))
1312rabbidv 3381 . . . . . . 7 (𝑝 = 𝑃 → {𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)} = {𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
1413fveqeq2d 6682 . . . . . 6 (𝑝 = 𝑃 → ((♯‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1514rspccv 3523 . . . . 5 (∀𝑝𝑉 (♯‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 → (𝑃𝑉 → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
16153ad2ant3 1136 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (♯‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾) → (𝑃𝑉 → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1710, 16syl 17 . . 3 (𝐺 RegUSGraph 𝐾 → (𝑃𝑉 → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1817imp 410 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾)
199, 18eqtrd 2773 1 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wral 3053  {crab 3057  Vcvv 3398  {cpr 4518   class class class wbr 5030  1-1-ontowf1o 6338  cfv 6339  0cc0 10615  1c1 10616  2c2 11771  0*cxnn0 12048  chash 13782  Word cword 13955  Vtxcvtx 26941  Edgcedg 26992  USGraphcusgr 27094   RegUSGraph crusgr 27498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-oadd 8135  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-dju 9403  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-n0 11977  df-xnn0 12049  df-z 12063  df-uz 12325  df-xadd 12591  df-fz 12982  df-fzo 13125  df-hash 13783  df-word 13956  df-edg 26993  df-uhgr 27003  df-ushgr 27004  df-upgr 27027  df-umgr 27028  df-uspgr 27095  df-usgr 27096  df-nbgr 27275  df-vtxdg 27408  df-rgr 27499  df-rusgr 27500
This theorem is referenced by:  rusgrnumwwlkl1  27906  clwwlknon2num  28042
  Copyright terms: Public domain W3C validator