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Theorem rusgrnumwrdl2 26352
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 6158 . . . . . 6 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2694 . . . . 5 𝑉 ∈ V
43wrdexi 13256 . . . 4 Word 𝑉 ∈ V
54rabex 4773 . . 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 13637 . . . 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 488 . . 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 13080 . . 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 68 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
111rusgrpropadjvtx 26351 . . . 4 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
12 preq1 4238 . . . . . . . . . 10 (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠})
1312eleq1d 2683 . . . . . . . . 9 (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ (Edg‘𝐺) ↔ {𝑃, 𝑠} ∈ (Edg‘𝐺)))
1413rabbidv 3177 . . . . . . . 8 (𝑝 = 𝑃 → {𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)} = {𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)})
1514fveq2d 6152 . . . . . . 7 (𝑝 = 𝑃 → (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}))
1615eqeq1d 2623 . . . . . 6 (𝑝 = 𝑃 → ((#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
1716rspccv 3292 . . . . 5 (∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾 → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾))
18173ad2ant3 1082 . . . 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 445 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ (Edg‘𝐺)}) = 𝐾)
2110, 20eqtrd 2655 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  {cpr 4150   class class class wbr 4613  1-1-ontowf1o 5846  cfv 5847  0cc0 9880  1c1 9881  2c2 11014  0*cxnn0 11307  #chash 13057  Word cword 13230  Vtxcvtx 25774  Edgcedg 25839   USGraph cusgr 25937   RegUSGraph crusgr 26322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-xadd 11891  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-edg 25840  df-uhgr 25849  df-ushgr 25850  df-upgr 25873  df-umgr 25874  df-uspgr 25938  df-usgr 25939  df-nbgr 26115  df-vtxdg 26249  df-rgr 26323  df-rusgr 26324
This theorem is referenced by:  rusgrnumwwlkl1  26730  numclwwlkovf2num  27074
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