Theorem uhgrf 29097

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 29095	. 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 1537   ∈ wcel 2108   ∖ cdif 3973  ∅c0 4352  𝒫 cpw 4622  {csn 4648  dom cdm 5700  ⟶wf 6569  ‘cfv 6573  Vtxcvtx 29031  iEdgciedg 29032  UHGraphcuhgr 29091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-uhgr 29093

This theorem is referenced by:  uhgrss  29099  uhgrfun  29101  uhgrn0  29102  uhgr0vb  29107  uhgrun  29109  uhgredgn0  29163  2pthon3v  29976  isubgrvtxuhgr  47736  isubgruhgr  47738  isubgr0uhgr  47743  uhgrimisgrgric  47783