Theorem uhgrfun 29123

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 29119	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6664	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887  ∅c0 4274  𝒫 cpw 4542  {csn 4568  dom cdm 5622  Fun wfun 6484  ‘cfv 6490  Vtxcvtx 29053  iEdgciedg 29054  UHGraphcuhgr 29113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-uhgr 29115

This theorem is referenced by:  lpvtx  29125  upgrle2  29162  uhgredgiedgb  29183  uhgriedg0edg0  29184  uhgrvtxedgiedgb  29193  edglnl  29200  numedglnl  29201  uhgr2edg  29265  ushgredgedg  29286  ushgredgedgloop  29288  0uhgrsubgr  29336  uhgrsubgrself  29337  subgruhgrfun  29339  subgruhgredgd  29341  subumgredg2  29342  subupgr  29344  uhgrspansubgrlem  29347  uhgrspansubgr  29348  uhgrspan1  29360  upgrreslem  29361  umgrreslem  29362  upgrres  29363  umgrres  29364  vtxduhgr0e  29536  vtxduhgrun  29541  vtxduhgrfiun  29542  finsumvtxdg2ssteplem1  29603  upgrewlkle2  29664  upgredginwlk  29693  wlkiswwlks1  29924  wlkiswwlksupgr2  29934  usgrwwlks2on  30015  umgrwwlks2on  30016  vdn0conngrumgrv2  30255  eulerpathpr  30299  eulercrct  30301  lfuhgr  35306  loop1cycl  35325  umgr2cycllem  35328  isubgrvtxuhgr  48298  isubgredg  48300  isubgrsubgr  48303  isubgr0uhgr  48307  uhgrimedgi  48324  isuspgrim0lem  48327  isuspgrim0  48328  upgrimwlklem2  48332  upgrimwlklem3  48333  upgrimtrlslem1  48338  clnbgrgrimlem  48367  clnbgrgrim  48368  grimedg  48369