Theorem ushgrf 29698

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 29696	. 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸⟶1-1(𝒫 𝑉 ∖ {∅})))
4	3	ibi 270	1 ⊢ (𝐺 ∈ USHGraph → 𝐸:dom 𝐸⟶1-1(𝒫 𝑉 ∖ {∅}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  {csn 4588  dom cdm 5658  ⟶1-1wf1 6563  ‘cfv 6566  Vtxcvtx 29631  iEdgciedg 29632  USHGraphcushgr 29692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fv 6574  df-ushgr 29694

This theorem is used by:  ushgruhgr  29704  uspgrupgrushgr  29817  ushgredgedg  29867  ushgredgedgloop  29869  gricushgr  49288  ushggricedg  49298