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| Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| isrgr.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| isrgr.d | ⊢ 𝐷 = (VtxDeg‘𝐺) | 
| Ref | Expression | 
|---|---|
| rgrprop | ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rgr 29575 | . . 3 ⊢ RegGraph = {〈𝑔, 𝑘〉 ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)} | |
| 2 | 1 | bropaex12 5777 | . 2 ⊢ (𝐺 RegGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V)) | 
| 3 | isrgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 4 | isrgr.d | . . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
| 5 | 3, 4 | isrgr 29577 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| 6 | 5 | biimpd 229 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| 7 | 2, 6 | mpcom 38 | 1 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 ℕ0*cxnn0 12599 Vtxcvtx 29013 VtxDegcvtxdg 29483 RegGraph crgr 29573 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-iota 6514 df-fv 6569 df-rgr 29575 | 
| This theorem is referenced by: rusgrprop0 29585 uhgr0edg0rgrb 29592 frrusgrord 30360 | 
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