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Theorem isrgr 26342
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 2686 . . . . 5 (𝑘 = 𝐾 → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
21adantl 482 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
3 fveq2 6153 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantr 481 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5 fveq2 6153 . . . . . . . 8 (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
65fveq1d 6155 . . . . . . 7 (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
76adantr 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8 simpr 477 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → 𝑘 = 𝐾)
97, 8eqeq12d 2636 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
104, 9raleqbidv 3144 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
112, 10anbi12d 746 . . 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 26340 . . 3 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
1311, 12brabga 4954 . 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 6154 . . . . . . 7 (𝐷𝑣) = ((VtxDeg‘𝐺)‘𝑣)
1716eqeq1i 2626 . . . . . 6 ((𝐷𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1814, 17raleqbii 2985 . . . . 5 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1918bicomi 214 . . . 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 739 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
2213, 21bitrd 268 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618  cfv 5852  0*cxnn0 11315  Vtxcvtx 25791  VtxDegcvtxdg 26265   RegGraph crgr 26338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-iota 5815  df-fv 5860  df-rgr 26340
This theorem is referenced by:  rgrprop  26343  isrusgr0  26349  0edg0rgr  26355  0vtxrgr  26359  rgrprcx  26375  frgrregorufrg  27080
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