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Theorem rgrprop 26587
 Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrgr.v 𝑉 = (Vtx‘𝐺)
isrgr.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
rgrprop (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)

Proof of Theorem rgrprop
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rgr 26584 . . . 4 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
21breqi 4766 . . 3 (𝐺RegGraph𝐾𝐺{⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}𝐾)
3 brabv 6816 . . 3 (𝐺{⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
42, 3sylbi 207 . 2 (𝐺RegGraph𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
5 isrgr.v . . . 4 𝑉 = (Vtx‘𝐺)
6 isrgr.d . . . 4 𝐷 = (VtxDeg‘𝐺)
75, 6isrgr 26586 . . 3 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
87biimpd 219 . 2 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
94, 8mpcom 38 1 (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1596   ∈ wcel 2103  ∀wral 3014  Vcvv 3304   class class class wbr 4760  {copab 4820  ‘cfv 6001  ℕ0*cxnn0 11476  Vtxcvtx 25994  VtxDegcvtxdg 26492  RegGraphcrgr 26582 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-iota 5964  df-fv 6009  df-rgr 26584 This theorem is referenced by:  rusgrprop0  26594  uhgr0edg0rgrb  26601  frrusgrord  27416
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