Theorem uhgrfun 29083

Description: The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 15-Dec-2020.)

Hypothesis

Ref	Expression
uhgrfun.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
uhgrfun	⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Proof of Theorem uhgrfun

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 29079	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6740	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948  ∅c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  Fun wfun 6555  ‘cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-uhgr 29075

This theorem is referenced by:  lpvtx  29085  upgrle2  29122  uhgredgiedgb  29143  uhgriedg0edg0  29144  uhgrvtxedgiedgb  29153  edglnl  29160  numedglnl  29161  uhgr2edg  29225  ushgredgedg  29246  ushgredgedgloop  29248  0uhgrsubgr  29296  uhgrsubgrself  29297  subgruhgrfun  29299  subgruhgredgd  29301  subumgredg2  29302  subupgr  29304  uhgrspansubgrlem  29307  uhgrspansubgr  29308  uhgrspan1  29320  upgrreslem  29321  umgrreslem  29322  upgrres  29323  umgrres  29324  vtxduhgr0e  29496  vtxduhgrun  29501  vtxduhgrfiun  29502  finsumvtxdg2ssteplem1  29563  upgrewlkle2  29624  upgredginwlk  29654  wlkiswwlks1  29887  wlkiswwlksupgr2  29897  umgrwwlks2on  29977  vdn0conngrumgrv2  30215  eulerpathpr  30259  eulercrct  30261  lfuhgr  35123  loop1cycl  35142  umgr2cycllem  35145  isubgrvtxuhgr  47850  isubgredg  47852  isubgrsubgr  47855  isubgr0uhgr  47859  isuspgrim0lem  47871  isuspgrim0  47872  clnbgrgrimlem  47901  clnbgrgrim  47902  grimedg  47903