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Theorem rusgranumwwlkl1 26211
Description: In a k-regular graph, the number of walks of length 1 represented as words (thus the number of edges) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Proof shortened by AV, 4-May-2021.)
Assertion
Ref Expression
rusgranumwwlkl1 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾)
Distinct variable groups:   𝑤,𝐸   𝑤,𝑃   𝑤,𝑉
Allowed substitution hint:   𝐾(𝑤)

Proof of Theorem rusgranumwwlkl1
Dummy variables 𝑓 𝑝 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusisusgra 26196 . . . . . . 7 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
2 usgrav 25605 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
32simpld 473 . . . . . . 7 (𝑉 USGrph 𝐸𝑉 ∈ V)
41, 3syl 17 . . . . . 6 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ V)
5 wrdexg 13029 . . . . . 6 (𝑉 ∈ V → Word 𝑉 ∈ V)
64, 5syl 17 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → Word 𝑉 ∈ V)
76adantr 479 . . . 4 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → Word 𝑉 ∈ V)
8 rabexg 4638 . . . 4 (Word 𝑉 ∈ V → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)} ∈ V)
97, 8syl 17 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)} ∈ V)
10 wrd2f1tovbij 13410 . . . 4 ((𝑉 ∈ V ∧ 𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}–1-1-onto→{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸})
114, 10sylan 486 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}–1-1-onto→{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸})
12 hasheqf1oi 12867 . . 3 ({𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)} ∈ V → (∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}–1-1-onto→{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸} → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸})))
139, 11, 12sylc 62 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}))
14 rusgraprop3 26208 . . . 4 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾))
15 preq1 4115 . . . . . . . . . 10 (𝑝 = 𝑃 → {𝑝, 𝑠} = {𝑃, 𝑠})
1615eleq1d 2576 . . . . . . . . 9 (𝑝 = 𝑃 → ({𝑝, 𝑠} ∈ ran 𝐸 ↔ {𝑃, 𝑠} ∈ ran 𝐸))
1716rabbidv 3068 . . . . . . . 8 (𝑝 = 𝑃 → {𝑠𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸} = {𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸})
1817fveq2d 5991 . . . . . . 7 (𝑝 = 𝑃 → (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}))
1918eqeq1d 2516 . . . . . 6 (𝑝 = 𝑃 → ((#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾))
2019rspccv 3183 . . . . 5 (∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾 → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾))
21203ad2ant3 1076 . . . 4 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 (#‘{𝑠𝑉 ∣ {𝑝, 𝑠} ∈ ran 𝐸}) = 𝐾) → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾))
2214, 21syl 17 . . 3 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑃𝑉 → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾))
2322imp 443 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑠𝑉 ∣ {𝑃, 𝑠} ∈ ran 𝐸}) = 𝐾)
2413, 23eqtrd 2548 1 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1938  wral 2800  {crab 2804  Vcvv 3077  {cpr 4030  cop 4034   class class class wbr 4481  ran crn 4933  1-1-ontowf1o 5688  cfv 5689  0cc0 9691  1c1 9692  2c2 10825  0cn0 11047  #chash 12847  Word cword 13005   USGrph cusg 25597   RegUSGrph crusgra 26188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-om 6834  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-1o 7323  df-2o 7324  df-oadd 7327  df-er 7505  df-map 7622  df-pm 7623  df-en 7718  df-dom 7719  df-sdom 7720  df-fin 7721  df-card 8524  df-cda 8749  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-2 10834  df-n0 11048  df-z 11119  df-uz 11428  df-xadd 11689  df-fz 12066  df-fzo 12203  df-hash 12848  df-word 13013  df-usgra 25600  df-nbgra 25687  df-vdgr 26159  df-rgra 26189  df-rusgra 26190
This theorem is referenced by:  rusgranumwlkl1  26212  numclwwlkovf2num  26350
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