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| Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| isrusgr0.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| isrusgr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) | 
| Ref | Expression | 
|---|---|
| isrusgr0 | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isrusgr 29579 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))) | |
| 2 | isrusgr0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | isrusgr0.d | . . . . 5 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
| 4 | 2, 3 | isrgr 29577 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| 5 | 4 | anbi2d 630 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))) | 
| 6 | 3anass 1095 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | |
| 7 | 5, 6 | bitr4di 289 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| 8 | 1, 7 | bitrd 279 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 ℕ0*cxnn0 12599 Vtxcvtx 29013 USGraphcusgr 29166 VtxDegcvtxdg 29483 RegGraph crgr 29573 RegUSGraph crusgr 29574 | 
| 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-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-iota 6514 df-fv 6569 df-rgr 29575 df-rusgr 29576 | 
| This theorem is referenced by: usgreqdrusgr 29586 cusgrrusgr 29599 rgrusgrprc 29607 rusgrprc 29608 | 
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