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Mirrors > Home > MPE Home > Th. List > ushgrf | Structured version Visualization version GIF version |
Description: The edge function of an undirected simple hypergraph is a one-to-one function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
uhgrf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrf.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
ushgrf | ⊢ (𝐺 ∈ USHGraph → 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrf.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgrf.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | isushgr 26413 | . 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅}))) |
4 | 3 | ibi 259 | 1 ⊢ (𝐺 ∈ USHGraph → 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ∅c0 4141 𝒫 cpw 4379 {csn 4398 dom cdm 5357 –1-1→wf1 6134 ‘cfv 6137 Vtxcvtx 26348 iEdgciedg 26349 USHGraphcushgr 26409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fv 6145 df-ushgr 26411 |
This theorem is referenced by: ushgruhgr 26421 uspgrupgrushgr 26530 ushgredgedg 26579 ushgredgedgloop 26581 ushgredgedgloopOLD 26582 isomushgr 42749 ushrisomgr 42764 |
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