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Theorem isrgr 26690
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) → (𝐺RegGraph𝐾 ↔ (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 2838 . . . . 5 (𝑘 = 𝐾 → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
21adantl 467 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
3 fveq2 6333 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantr 466 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5 fveq2 6333 . . . . . . . 8 (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
65fveq1d 6335 . . . . . . 7 (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
76adantr 466 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8 simpr 471 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → 𝑘 = 𝐾)
97, 8eqeq12d 2786 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
104, 9raleqbidv 3301 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
112, 10anbi12d 616 . . 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 26688 . . 3 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
1311, 12brabga 5123 . 2 ((𝐺𝑊𝐾𝑍) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)))
14 isrgr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
15 isrgr.d . . . . . . . 8 𝐷 = (VtxDeg‘𝐺)
1615fveq1i 6334 . . . . . . 7 (𝐷𝑣) = ((VtxDeg‘𝐺)‘𝑣)
1716eqeq1i 2776 . . . . . 6 ((𝐷𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1814, 17raleqbii 3139 . . . . 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 614 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
2213, 21bitrd 268 1 ((𝐺𝑊𝐾𝑍) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4787  cfv 6030  0*cxnn0 11570  Vtxcvtx 26095  VtxDegcvtxdg 26596  RegGraphcrgr 26686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-iota 5993  df-fv 6038  df-rgr 26688
This theorem is referenced by:  rgrprop  26691  isrusgr0  26697  0edg0rgr  26703  0vtxrgr  26707  rgrprcx  26723
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