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Theorem isrgr 27937
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 2828 . . . . 5 (𝑘 = 𝐾 → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
21adantl 482 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
3 fveq2 6771 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantr 481 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5 fveq2 6771 . . . . . . . 8 (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
65fveq1d 6773 . . . . . . 7 (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
76adantr 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8 simpr 485 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → 𝑘 = 𝐾)
97, 8eqeq12d 2756 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
104, 9raleqbidv 3335 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
112, 10anbi12d 631 . . 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 27935 . . 3 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
1311, 12brabga 5450 . 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 6772 . . . . . . 7 (𝐷𝑣) = ((VtxDeg‘𝐺)‘𝑣)
1716eqeq1i 2745 . . . . . 6 ((𝐷𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1814, 17raleqbii 3163 . . . . 5 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1918bicomi 223 . . . 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 629 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
2213, 21bitrd 278 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wcel 2110  wral 3066   class class class wbr 5079  cfv 6432  0*cxnn0 12316  Vtxcvtx 27377  VtxDegcvtxdg 27843   RegGraph crgr 27933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-iota 6390  df-fv 6440  df-rgr 27935
This theorem is referenced by:  rgrprop  27938  isrusgr0  27944  0edg0rgr  27950  0vtxrgr  27954  rgrprcx  27970
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