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Theorem frgraregorufrg 26393
Description: If there is a vertex having degree 𝑘 for each nonnegative integer 𝑘 in a friendship graph, then there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". Variant of frgraregorufr 26374 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
Assertion
Ref Expression
frgraregorufrg (𝑉 FriendGrph 𝐸 → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝑘 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
Distinct variable groups:   𝐸,𝑎,𝑘,𝑣,𝑤   𝑉,𝑎,𝑘,𝑣,𝑤

Proof of Theorem frgraregorufrg
StepHypRef Expression
1 frgraregorufr 26374 . . . 4 (𝑉 FriendGrph 𝐸 → (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
21adantr 479 . . 3 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
3 frisusgra 26313 . . . . . . . . 9 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
43anim1i 589 . . . . . . . 8 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → (𝑉 USGrph 𝐸𝑘 ∈ ℕ0))
54anim1i 589 . . . . . . 7 (((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘) → ((𝑉 USGrph 𝐸𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘))
6 df-3an 1032 . . . . . . 7 ((𝑉 USGrph 𝐸𝑘 ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘) ↔ ((𝑉 USGrph 𝐸𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘))
75, 6sylibr 222 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘) → (𝑉 USGrph 𝐸𝑘 ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘))
8 usgrav 25661 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
93, 8syl 17 . . . . . . . . . 10 (𝑉 FriendGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
109anim1i 589 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑘 ∈ ℕ0))
11 df-3an 1032 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑘 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑘 ∈ ℕ0))
1210, 11sylibr 222 . . . . . . . 8 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑘 ∈ ℕ0))
13 isrusgra 26248 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑘 ∈ ℕ0) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ↔ (𝑉 USGrph 𝐸𝑘 ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘)))
1412, 13syl 17 . . . . . . 7 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ↔ (𝑉 USGrph 𝐸𝑘 ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘)))
1514adantr 479 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ↔ (𝑉 USGrph 𝐸𝑘 ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘)))
167, 15mpbird 245 . . . . 5 (((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝑘)
1716ex 448 . . . 4 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘 → ⟨𝑉, 𝐸⟩ RegUSGrph 𝑘))
1817orim1d 879 . . 3 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → ((∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
192, 18syld 45 . 2 ((𝑉 FriendGrph 𝐸𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝑘 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
2019ralrimiva 2948 1 (𝑉 FriendGrph 𝐸 → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝑘 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  {csn 4124  {cpr 4126  cop 4130   class class class wbr 4577  ran crn 5029  cfv 5790  (class class class)co 6527  0cn0 11142   USGrph cusg 25653   VDeg cvdg 26214   RegUSGrph crusgra 26244   FriendGrph cfrgra 26309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-xadd 11782  df-fz 12156  df-hash 12938  df-usgra 25656  df-nbgra 25743  df-vdgr 26215  df-rgra 26245  df-rusgra 26246  df-frgra 26310
This theorem is referenced by:  friendshipgt3  26442
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