Theorem uhgrfun 28326

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 2733	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 28322	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6722	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ∖ cdif 3946  ∅c0 4323  𝒫 cpw 4603  {csn 4629  dom cdm 5677  Fun wfun 6538  ‘cfv 6544  Vtxcvtx 28256  iEdgciedg 28257  UHGraphcuhgr 28316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-uhgr 28318

This theorem is referenced by:  lpvtx  28328  upgrle2  28365  uhgredgiedgb  28386  uhgriedg0edg0  28387  uhgrvtxedgiedgb  28396  edglnl  28403  numedglnl  28404  uhgr2edg  28465  ushgredgedg  28486  ushgredgedgloop  28488  0uhgrsubgr  28536  uhgrsubgrself  28537  subgruhgrfun  28539  subgruhgredgd  28541  subumgredg2  28542  subupgr  28544  uhgrspansubgrlem  28547  uhgrspansubgr  28548  uhgrspan1  28560  upgrreslem  28561  umgrreslem  28562  upgrres  28563  umgrres  28564  vtxduhgr0e  28735  vtxduhgrun  28740  vtxduhgrfiun  28741  finsumvtxdg2ssteplem1  28802  upgrewlkle2  28863  upgredginwlk  28893  wlkiswwlks1  29121  wlkiswwlksupgr2  29131  umgrwwlks2on  29211  vdn0conngrumgrv2  29449  eulerpathpr  29493  eulercrct  29495  lfuhgr  34108  loop1cycl  34128  umgr2cycllem  34131