MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrf Structured version   Visualization version   GIF version

Theorem uhgrf 29080
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 29078 . 2 (𝐺 ∈ UHGraph → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
43ibi 267 1 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cdif 3947  c0 4332  𝒫 cpw 4599  {csn 4625  dom cdm 5684  wf 6556  cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  UHGraphcuhgr 29074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-uhgr 29076
This theorem is referenced by:  uhgrss  29082  uhgrfun  29084  uhgrn0  29085  uhgr0vb  29090  uhgrun  29092  uhgredgn0  29146  2pthon3v  29964  isubgrvtxuhgr  47855  isubgredg  47857  isubgruhgr  47859  isubgr0uhgr  47864  uhgrimisgrgric  47904
  Copyright terms: Public domain W3C validator