Theorem ushgrf 29647

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 29645	. 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 1569   ∈ wcel 2144   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4584  dom cdm 5650  –1-1→wf1 6536  ‘cfv 6539  Vtxcvtx 29580  iEdgciedg 29581  USHGraphcushgr 29641

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fv 6547  df-ushgr 29643

This theorem is used by:  ushgruhgr  29653  uspgrupgrushgr  29766  ushgredgedg  29816  ushgredgedgloop  29818  gricushgr  49109  ushggricedg  49119