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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 28012 | . 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅}))) |
4 | 3 | ibi 266 | 1 ⊢ (𝐺 ∈ USHGraph → 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∖ cdif 3907 ∅c0 4282 𝒫 cpw 4560 {csn 4586 dom cdm 5633 –1-1→wf1 6493 ‘cfv 6496 Vtxcvtx 27947 iEdgciedg 27948 USHGraphcushgr 28008 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5263 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fv 6504 df-ushgr 28010 |
This theorem is referenced by: ushgruhgr 28020 uspgrupgrushgr 28128 ushgredgedg 28177 ushgredgedgloop 28179 isomushgr 46008 ushrisomgr 46023 |
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