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Mirrors > Home > MPE Home > Th. List > Mathboxes > rusgrprop | Structured version Visualization version GIF version |
Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
Ref | Expression |
---|---|
rusgrprop | ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rusgr 40758 | . . . 4 ⊢ RegUSGraph = {〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)} | |
2 | 1 | breqi 4589 | . . 3 ⊢ (𝐺 RegUSGraph 𝐾 ↔ 𝐺{〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}𝐾) |
3 | brabv 6597 | . . 3 ⊢ (𝐺{〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V)) | |
4 | 2, 3 | sylbi 206 | . 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V)) |
5 | isrusgr 40761 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))) | |
6 | 5 | biimpd 218 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))) |
7 | 4, 6 | mpcom 37 | 1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 USGraph cusgr 40379 RegGraph crgr 40755 RegUSGraph crusgr 40756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rusgr 40758 |
This theorem is referenced by: rusgrrgr 40763 rusgrusgr 40764 rusgrprop0 40767 |
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