Theorem uhgrfun 29029

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 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 29025	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6660	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4553  {csn 4579  dom cdm 5623  Fun wfun 6480  ‘cfv 6486  Vtxcvtx 28959  iEdgciedg 28960  UHGraphcuhgr 29019

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-uhgr 29021

This theorem is referenced by:  lpvtx  29031  upgrle2  29068  uhgredgiedgb  29089  uhgriedg0edg0  29090  uhgrvtxedgiedgb  29099  edglnl  29106  numedglnl  29107  uhgr2edg  29171  ushgredgedg  29192  ushgredgedgloop  29194  0uhgrsubgr  29242  uhgrsubgrself  29243  subgruhgrfun  29245  subgruhgredgd  29247  subumgredg2  29248  subupgr  29250  uhgrspansubgrlem  29253  uhgrspansubgr  29254  uhgrspan1  29266  upgrreslem  29267  umgrreslem  29268  upgrres  29269  umgrres  29270  vtxduhgr0e  29442  vtxduhgrun  29447  vtxduhgrfiun  29448  finsumvtxdg2ssteplem1  29509  upgrewlkle2  29570  upgredginwlk  29599  wlkiswwlks1  29830  wlkiswwlksupgr2  29840  umgrwwlks2on  29920  vdn0conngrumgrv2  30158  eulerpathpr  30202  eulercrct  30204  lfuhgr  35093  loop1cycl  35112  umgr2cycllem  35115  isubgrvtxuhgr  47852  isubgredg  47854  isubgrsubgr  47857  isubgr0uhgr  47861  uhgrimedgi  47878  isuspgrim0lem  47881  isuspgrim0  47882  upgrimwlklem2  47886  upgrimwlklem3  47887  upgrimtrlslem1  47892  clnbgrgrimlem  47921  clnbgrgrim  47922  grimedg  47923