Theorem isrgr 29487

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	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 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.

Step	Hyp	Ref	Expression
1		eleq1 2816	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 ∈ ℕ0* ↔ 𝐾 ∈ ℕ0*))
2	1	adantl 481	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑘 ∈ ℕ0* ↔ 𝐾 ∈ ℕ0*))
3		fveq2 6858	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantr 480	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5		fveq2 6858	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
6	5	fveq1d 6860	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8		simpr 484	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
9	7, 8	eqeq12d 2745	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
10	4, 9	raleqbidv 3319	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
11	2, 10	anbi12d 632	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)))
12		df-rgr 29485	. . 3 ⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
13	11, 12	brabga 5494	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (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)
16	15	fveq1i 6859	. . . . . . 7 ⊢ (𝐷‘𝑣) = ((VtxDeg‘𝐺)‘𝑣)
17	16	eqeq1i 2734	. . . . . 6 ⊢ ((𝐷‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
18	14, 17	raleqbii 3317	. . . . 5 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
19	18	bicomi 224	. . . 4 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
20	19	a1i 11	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
21	20	anbi2d 630	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
22	13, 21	bitrd 279	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107  ‘cfv 6511  ℕ₀^*cxnn0 12515  Vtxcvtx 28923  VtxDegcvtxdg 29393   RegGraph crgr 29483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-iota 6464  df-fv 6519  df-rgr 29485

This theorem is referenced by:  rgrprop  29488  isrusgr0  29494  0edg0rgr  29500  0vtxrgr  29504  rgrprcx  29520