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Theorem ushgrf 28014
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.)
Hypotheses
Ref Expression
uhgrf.v 𝑉 = (Vtx‘𝐺)
uhgrf.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
ushgrf (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))

Proof of Theorem ushgrf
StepHypRef Expression
1 uhgrf.v . . 3 𝑉 = (Vtx‘𝐺)
2 uhgrf.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2isushgr 28012 . 2 (𝐺 ∈ USHGraph → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
43ibi 266 1 (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(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  1-1wf1 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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