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Theorem isrgr 26907
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 2847 . . . . 5 (𝑘 = 𝐾 → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
21adantl 475 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑘 ∈ ℕ0*𝐾 ∈ ℕ0*))
3 fveq2 6446 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantr 474 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5 fveq2 6446 . . . . . . . 8 (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
65fveq1d 6448 . . . . . . 7 (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
76adantr 474 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8 simpr 479 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝐾) → 𝑘 = 𝐾)
97, 8eqeq12d 2793 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
104, 9raleqbidv 3326 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
112, 10anbi12d 624 . . 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 26905 . . 3 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
1311, 12brabga 5226 . 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 6447 . . . . . . 7 (𝐷𝑣) = ((VtxDeg‘𝐺)‘𝑣)
1716eqeq1i 2783 . . . . . 6 ((𝐷𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1814, 17raleqbii 3172 . . . . 5 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
1918bicomi 216 . . . 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 622 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
2213, 21bitrd 271 1 ((𝐺𝑊𝐾𝑍) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090   class class class wbr 4886  cfv 6135  0*cxnn0 11714  Vtxcvtx 26344  VtxDegcvtxdg 26813  RegGraphcrgr 26903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-iota 6099  df-fv 6143  df-rgr 26905
This theorem is referenced by:  rgrprop  26908  isrusgr0  26914  0edg0rgr  26920  0vtxrgr  26924  rgrprcx  26940
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