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Theorem rusgrnumwrdl2 40786
 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)
2 fvex 6113 . . . . . 6 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2684 . . . . 5 𝑉 ∈ V
43wrdexi 13172 . . . 4 Word 𝑉 ∈ V
54rabex 4740 . . 3 {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))} ∈ V
63a1i 11 . . . 4 (𝐺 RegUSGraph 𝐾𝑉 ∈ V)
7 wrd2f1tovbij 13551 . . . 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)})
86, 7sylan 487 . . 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)})
9 hasheqf1oi 13002 . . 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)})))
105, 8, 9mpsyl 66 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
111rusgrpropadjvtx 40785 . . . 4 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
12 preq1 4212 . . . . . . . . . 10 (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠})
1312eleq1d 2672 . . . . . . . . 9 (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ (Edg‘𝐺) ↔ {𝑃, 𝑠} ∈ (Edg‘𝐺)))
1413rabbidv 3164 . . . . . . . 8 (𝑝 = 𝑃 → {𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)} = {𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
1514fveq2d 6107 . . . . . . 7 (𝑝 = 𝑃 → (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
1615eqeq1d 2612 . . . . . 6 (𝑝 = 𝑃 → ((#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1716rspccv 3279 . . . . 5 (∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
18173ad2ant3 1077 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾) → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1911, 18syl 17 . . 3 (𝐺 RegUSGraph 𝐾 → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
2019imp 444 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾)
2110, 20eqtrd 2644 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 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  {cpr 4127   class class class wbr 4583  –1-1-onto→wf1o 5803  ‘cfv 5804  0cc0 9815  1c1 9816  2c2 10947  ℕ0*cxnn0 11240  #chash 12979  Word cword 13146  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   RegUSGraph crusgr 40756 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-nbgr 40554  df-vtxdg 40682  df-rgr 40757  df-rusgr 40758 This theorem is referenced by:  rusgrnumwwlkl1  41172  av-numclwwlkovf2num  41516
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