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Mirrors > Home > MPE Home > Th. List > isrusgr | Structured version Visualization version GIF version |
Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
Ref | Expression |
---|---|
isrusgr | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2902 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph)) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph)) |
3 | breq12 5073 | . . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 RegGraph 𝑘 ↔ 𝐺 RegGraph 𝐾)) | |
4 | 2, 3 | anbi12d 632 | . 2 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘) ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))) |
5 | df-rusgr 27342 | . 2 ⊢ RegUSGraph = {〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)} | |
6 | 4, 5 | brabga 5423 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 USGraphcusgr 26936 RegGraph crgr 27339 RegUSGraph crusgr 27340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-rusgr 27342 |
This theorem is referenced by: rusgrprop 27346 isrusgr0 27350 usgr0edg0rusgr 27359 0vtxrusgr 27361 |
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