Theorem ushgrf 28990

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 𝐸–1-1→(𝒫 𝑉 ∖ {∅}))

Proof of Theorem ushgrf

Step	Hyp	Ref	Expression
1		uhgrf.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrf.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isushgr 28988	. 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})))
4	3	ibi 267	1 ⊢ (𝐺 ∈ USHGraph → 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911  ∅c0 4296  𝒫 cpw 4563  {csn 4589  dom cdm 5638  –1-1→wf1 6508  ‘cfv 6511  Vtxcvtx 28923  iEdgciedg 28924  USHGraphcushgr 28984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fv 6519  df-ushgr 28986

This theorem is referenced by:  ushgruhgr  28996  uspgrupgrushgr  29106  ushgredgedg  29156  ushgredgedgloop  29158  gricushgr  47914  ushggricedg  47924