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Theorem rusgrnumwrdl2 26891
 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 ((𝐺RegUSGraph𝐾𝑃𝑉) → (♯‘{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 6451 . . . . 5 𝑉 ∈ V
32wrdexi 13593 . . . 4 Word 𝑉 ∈ V
43rabex 5039 . . 3 {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))} ∈ V
52a1i 11 . . . 4 (𝐺RegUSGraph𝐾𝑉 ∈ V)
6 wrd2f1tovbij 14089 . . . 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 575 . . 3 ((𝐺RegUSGraph𝐾𝑃𝑉) → ∃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 13439 . . 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 ((𝐺RegUSGraph𝐾𝑃𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
101rusgrpropadjvtx 26890 . . . 4 (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (♯‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
11 preq1 4488 . . . . . . . . 9 (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠})
1211eleq1d 2891 . . . . . . . 8 (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ (Edg‘𝐺) ↔ {𝑃, 𝑠} ∈ (Edg‘𝐺)))
1312rabbidv 3402 . . . . . . 7 (𝑝 = 𝑃 → {𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)} = {𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
1413fveqeq2d 6445 . . . . . 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 1169 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (♯‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾) → (𝑃𝑉 → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1710, 16syl 17 . . 3 (𝐺RegUSGraph𝐾 → (𝑃𝑉 → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1817imp 397 . 2 ((𝐺RegUSGraph𝐾𝑃𝑉) → (♯‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾)
199, 18eqtrd 2861 1 ((𝐺RegUSGraph𝐾𝑃𝑉) → (♯‘{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 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∀wral 3117  {crab 3121  Vcvv 3414  {cpr 4401   class class class wbr 4875  –1-1-onto→wf1o 6126  ‘cfv 6127  0cc0 10259  1c1 10260  2c2 11413  ℕ0*cxnn0 11697  ♯chash 13417  Word cword 13581  Vtxcvtx 26301  Edgcedg 26352  USGraphcusgr 26455  RegUSGraphcrusgr 26861 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-xadd 12240  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-edg 26353  df-uhgr 26363  df-ushgr 26364  df-upgr 26387  df-umgr 26388  df-uspgr 26456  df-usgr 26457  df-nbgr 26637  df-vtxdg 26771  df-rgr 26862  df-rusgr 26863 This theorem is referenced by:  rusgrnumwwlkl1  27304  clwwlknon2num  27476
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