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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 4932 | . . 3 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | eqid 2622 | . . . 4 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
3 | eqid 2622 | . . . 4 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | 2, 3 | isuspgr 26047 | . . 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 25887 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
7 | 6 | dmeqd 5326 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘〈𝑉, 𝐸〉) = dom 𝐸) |
8 | opvtxfv 25884 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
9 | 8 | pweqd 4163 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘〈𝑉, 𝐸〉) = 𝒫 𝑉) |
10 | 9 | difeq1d 3727 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 10 | rabeqdv 3194 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {𝑝 ∈ (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ∣ (#‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}) |
12 | 6, 7, 11 | f1eq123d 6131 | . 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 268 | 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 196 ∧ wa 384 ∈ wcel 1990 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∅c0 3915 𝒫 cpw 4158 {csn 4177 〈cop 4183 class class class wbr 4653 dom cdm 5114 –1-1→wf1 5885 ‘cfv 5888 ≤ cle 10075 2c2 11070 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 USPGraph cuspgr 26043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fv 5896 df-1st 7168 df-2nd 7169 df-vtx 25876 df-iedg 25877 df-uspgr 26045 |
This theorem is referenced by: uspgrsprfo 41756 |
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