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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 5358 | . . 3 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
2 | eqid 2823 | . . . 4 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
3 | eqid 2823 | . . . 4 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
4 | 2, 3 | isuhgr 26847 | . . 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 26794 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
7 | 6 | dmeqd 5776 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘〈𝑉, 𝐸〉) = dom 𝐸) |
8 | opvtxfv 26791 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
9 | 8 | pweqd 4560 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘〈𝑉, 𝐸〉) = 𝒫 𝑉) |
10 | 9 | difeq1d 4100 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘〈𝑉, 𝐸〉) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 6, 7, 10 | feq123d 6505 | . 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 281 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∅c0 4293 𝒫 cpw 4541 {csn 4569 〈cop 4575 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-1st 7691 df-2nd 7692 df-vtx 26785 df-iedg 26786 df-uhgr 26845 |
This theorem is referenced by: pliguhgr 28265 |
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