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Theorem isrgr 40759
 Description: The property of a class being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrgr.v 𝑉 = (Vtx‘𝐺)
isrgr.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
isrgr ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)   𝑊(𝑣)   𝑍(𝑣)

Proof of Theorem isrgr
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2676 . . . . 5 (𝑘 = 𝐾 → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
21adantl 481 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
3 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantr 480 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5 fveq2 6103 . . . . . . . 8 (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
65fveq1d 6105 . . . . . . 7 (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
76adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8 simpr 476 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → 𝑘 = 𝐾)
97, 8eqeq12d 2625 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
104, 9raleqbidv 3129 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
112, 10anbi12d 743 . . 3 ((𝑔 = 𝐺𝑘 = 𝐾) → ((𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)))
12 df-rgr 40757 . . 3 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
1311, 12brabga 4914 . 2 ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)))
14 isrgr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
15 isrgr.d . . . . . . . 8 𝐷 = (VtxDeg‘𝐺)
1615fveq1i 6104 . . . . . . 7 (𝐷𝑣) = ((VtxDeg‘𝐺)‘𝑣)
1716eqeq1i 2615 . . . . . 6 ((𝐷𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1814, 17raleqbii 2973 . . . . 5 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1918bicomi 213 . . . 4 (∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)
2019a1i 11 . . 3 ((𝐺𝑊𝐾𝑍) → (∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
2120anbi2d 736 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
2213, 21bitrd 267 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  ℕ0*cxnn0 11240  Vtxcvtx 25673  VtxDegcvtxdg 40681   RegGraph crgr 40755 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-rgr 40757 This theorem is referenced by:  rgrprop  40760  isrusgr0  40766  0edg0rgr  40772  0vtxrgr  40776  rgrprcx  40792  frgrregorufrg  41505
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