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	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺RegGraph𝐾 ↔ (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 2847	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 ∈ ℕ0* ↔ 𝐾 ∈ ℕ0*))
2	1	adantl 475	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑘 ∈ ℕ0* ↔ 𝐾 ∈ ℕ0*))
3		fveq2 6446	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	adantr 474	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (Vtx‘𝑔) = (Vtx‘𝐺))
5		fveq2 6446	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (VtxDeg‘𝑔) = (VtxDeg‘𝐺))
6	5	fveq1d 6448	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
7	6	adantr 474	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((VtxDeg‘𝑔)‘𝑣) = ((VtxDeg‘𝐺)‘𝑣))
8		simpr 479	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
9	7, 8	eqeq12d 2793	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
10	4, 9	raleqbidv 3326	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾))
11	2, 10	anbi12d 624	. . 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 26905	. . 3 ⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
13	11, 12	brabga 5226	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺RegGraph𝐾 ↔ (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 6447	. . . . . . 7 ⊢ (𝐷‘𝑣) = ((VtxDeg‘𝐺)‘𝑣)
17	16	eqeq1i 2783	. . . . . 6 ⊢ ((𝐷‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
18	14, 17	raleqbii 3172	. . . . 5 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝐾)
19	18	bicomi 216	. . . 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 622	. 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 271	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   class class class wbr 4886  ‘cfv 6135  ℕ₀^*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