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Mirrors > Home > MPE Home > Th. List > isuhgrop | Structured version Visualization version GIF version |
Description: The property of being an undirected hypergraph 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, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
isuhgrop | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5333 | . . 3 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | eqid 2736 | . . . 4 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
3 | eqid 2736 | . . . 4 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | 2, 3 | isuhgr 27105 | . . 3 ⊢ (〈𝑉, 𝐸〉 ∈ V → (〈𝑉, 𝐸〉 ∈ UHGraph ↔ (iEdg‘〈𝑉, 𝐸〉):dom (iEdg‘〈𝑉, 𝐸〉)⟶(𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}))) |
5 | 1, 4 | mp1i 13 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ UHGraph ↔ (iEdg‘〈𝑉, 𝐸〉):dom (iEdg‘〈𝑉, 𝐸〉)⟶(𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}))) |
6 | opiedgfv 27052 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
7 | 6 | dmeqd 5759 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘〈𝑉, 𝐸〉) = dom 𝐸) |
8 | opvtxfv 27049 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
9 | 8 | pweqd 4518 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘〈𝑉, 𝐸〉) = 𝒫 𝑉) |
10 | 9 | difeq1d 4022 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 6, 7, 10 | feq123d 6512 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ((iEdg‘〈𝑉, 𝐸〉):dom (iEdg‘〈𝑉, 𝐸〉)⟶(𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
12 | 5, 11 | bitrd 282 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 ∅c0 4223 𝒫 cpw 4499 {csn 4527 〈cop 4533 dom cdm 5536 ⟶wf 6354 ‘cfv 6358 Vtxcvtx 27041 iEdgciedg 27042 UHGraphcuhgr 27101 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-1st 7739 df-2nd 7740 df-vtx 27043 df-iedg 27044 df-uhgr 27103 |
This theorem is referenced by: pliguhgr 28521 |
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