Theorem uhgrf 28794

Description: The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)

Hypotheses

Ref	Expression
uhgrf.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrf.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
uhgrf	⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))

Proof of Theorem uhgrf

Step	Hyp	Ref	Expression
1		uhgrf.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrf.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isuhgr 28792	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4	3	ibi 267	1 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∖ cdif 3938  ∅c0 4315  𝒫 cpw 4595  {csn 4621  dom cdm 5667  ⟶wf 6530  ‘cfv 6534  Vtxcvtx 28728  iEdgciedg 28729  UHGraphcuhgr 28788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-uhgr 28790

This theorem is referenced by:  uhgrss  28796  uhgrfun  28798  uhgrn0  28799  uhgr0vb  28804  uhgrun  28806  uhgredgn0  28860  2pthon3v  29669