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Theorem uhgrf 26367
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
StepHypRef Expression
1 uhgrf.v . . 3 𝑉 = (Vtx‘𝐺)
2 uhgrf.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2isuhgr 26365 . 2 (𝐺 ∈ UHGraph → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
43ibi 259 1 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cdif 3795  c0 4146  𝒫 cpw 4380  {csn 4399  dom cdm 5346  wf 6123  cfv 6127  Vtxcvtx 26301  iEdgciedg 26302  UHGraphcuhgr 26361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-uhgr 26363
This theorem is referenced by:  uhgrss  26369  uhgrfun  26371  uhgrn0  26372  uhgr0vb  26377  uhgrun  26379  uhgredgn0  26433  2pthon3v  27279
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