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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 5321 | . . 3 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | eqid 2798 | . . . 4 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
3 | eqid 2798 | . . . 4 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | 2, 3 | isuspgr 26945 | . . 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 26800 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
7 | 6 | dmeqd 5738 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘〈𝑉, 𝐸〉) = dom 𝐸) |
8 | opvtxfv 26797 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
9 | 8 | pweqd 4516 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘〈𝑉, 𝐸〉) = 𝒫 𝑉) |
10 | 9 | difeq1d 4049 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 10 | rabeqdv 3432 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {𝑝 ∈ (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) |
12 | 6, 7, 11 | f1eq123d 6583 | . 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 3110 Vcvv 3441 ∖ cdif 3878 ∅c0 4243 𝒫 cpw 4497 {csn 4525 〈cop 4531 class class class wbr 5030 dom cdm 5519 –1-1→wf1 6321 ‘cfv 6324 ≤ cle 10665 2c2 11680 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 USPGraphcuspgr 26941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fv 6332 df-1st 7671 df-2nd 7672 df-vtx 26791 df-iedg 26792 df-uspgr 26943 |
This theorem is referenced by: uspgrsprfo 44376 |
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