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Mirrors > Home > MPE Home > Th. List > isuspgrop | Structured version Visualization version GIF version |
Description: The property of being an undirected simple pseudograph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 25-Nov-2021.) |
Ref | Expression |
---|---|
isuspgrop | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5363 | . . 3 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | eqid 2738 | . . . 4 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
3 | eqid 2738 | . . . 4 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | 2, 3 | isuspgr 27267 | . . 3 ⊢ (〈𝑉, 𝐸〉 ∈ V → (〈𝑉, 𝐸〉 ∈ USPGraph ↔ (iEdg‘〈𝑉, 𝐸〉):dom (iEdg‘〈𝑉, 𝐸〉)–1-1→{𝑝 ∈ (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
5 | 1, 4 | mp1i 13 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ USPGraph ↔ (iEdg‘〈𝑉, 𝐸〉):dom (iEdg‘〈𝑉, 𝐸〉)–1-1→{𝑝 ∈ (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
6 | opiedgfv 27122 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
7 | 6 | dmeqd 5789 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘〈𝑉, 𝐸〉) = dom 𝐸) |
8 | opvtxfv 27119 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
9 | 8 | pweqd 4547 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘〈𝑉, 𝐸〉) = 𝒫 𝑉) |
10 | 9 | difeq1d 4051 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 10 | rabeqdv 3408 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {𝑝 ∈ (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) |
12 | 6, 7, 11 | f1eq123d 6672 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ((iEdg‘〈𝑉, 𝐸〉):dom (iEdg‘〈𝑉, 𝐸〉)–1-1→{𝑝 ∈ (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
13 | 5, 12 | bitrd 282 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 {crab 3066 Vcvv 3421 ∖ cdif 3878 ∅c0 4252 𝒫 cpw 4528 {csn 4556 〈cop 4562 class class class wbr 5068 dom cdm 5566 –1-1→wf1 6395 ‘cfv 6398 ≤ cle 10893 2c2 11910 ♯chash 13921 Vtxcvtx 27111 iEdgciedg 27112 USPGraphcuspgr 27263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fv 6406 df-1st 7780 df-2nd 7781 df-vtx 27113 df-iedg 27114 df-uspgr 27265 |
This theorem is referenced by: uspgrsprfo 45012 |
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