rgrprop - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rgrprop		Structured version Visualization version GIF version

Theorem rgrprop 26587

Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)

**Hypotheses**
Ref	Expression
isrgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isrgr.d	⊢ 𝐷 = (VtxDeg‘𝐺)

**Assertion**
Ref	Expression
rgrprop	⊢ (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Distinct variable groups: 𝑣,𝐺 𝑣,𝐾 Allowed substitution hints: 𝐷(𝑣) 𝑉(𝑣)

Proof of Theorem rgrprop Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rgr 26584	. . . 4 ⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ₀^* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
2	1	breqi 4766	. . 3 ⊢ (𝐺RegGraph𝐾 ↔ 𝐺{⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ₀^* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}𝐾)
3		brabv 6816	. . 3 ⊢ (𝐺{⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ₀^* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
4	2, 3	sylbi 207	. 2 ⊢ (𝐺RegGraph𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
5		isrgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
6		isrgr.d	. . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺)
7	5, 6	isrgr 26586	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
8	7	biimpd 219	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
9	4, 8	mpcom 38	1 ⊢ (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1596 ∈ wcel 2103 ∀wral 3014 Vcvv 3304 class class class wbr 4760 {copab 4820 ‘cfv 6001 ℕ₀^*cxnn0 11476 Vtxcvtx 25994 VtxDegcvtxdg 26492 RegGraphcrgr 26582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-sep 4889 ax-nul 4897 ax-pr 5011

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-ral 3019 df-rex 3020 df-rab 3023 df-v 3306 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-sn 4286 df-pr 4288 df-op 4292 df-uni 4545 df-br 4761 df-opab 4821 df-iota 5964 df-fv 6009 df-rgr 26584

This theorem is referenced by: rusgrprop0 26594 uhgr0edg0rgrb 26601 frrusgrord 27416

W3C validator