Theorem isrgr 29496

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 2814	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 ∈ ℕ0* ↔ 𝐾 ∈ ℕ0*))
2	1	adantl 480	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑘 ∈ ℕ0* ↔ 𝐾 ∈ ℕ0*))
3		fveq2 6901	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantr 479	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5		fveq2 6901	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
6	5	fveq1d 6903	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
7	6	adantr 479	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8		simpr 483	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
9	7, 8	eqeq12d 2742	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
10	4, 9	raleqbidv 3330	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
11	2, 10	anbi12d 630	. . 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 29494	. . 3 ⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
13	11, 12	brabga 5540	. 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 6902	. . . . . . 7 ⊢ (𝐷‘𝑣) = ((VtxDeg‘𝐺)‘𝑣)
17	16	eqeq1i 2731	. . . . . 6 ⊢ ((𝐷‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
18	14, 17	raleqbii 3328	. . . . 5 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
19	18	bicomi 223	. . . 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 628	. 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 278	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051   class class class wbr 5153  ‘cfv 6554  ℕ₀^*cxnn0 12596  Vtxcvtx 28932  VtxDegcvtxdg 29402   RegGraph crgr 29492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-iota 6506  df-fv 6562  df-rgr 29494

This theorem is referenced by:  rgrprop  29497  isrusgr0  29503  0edg0rgr  29509  0vtxrgr  29513  rgrprcx  29529