Theorem uhgrfun 28593

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 2730	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 28589	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6720	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104   ∖ cdif 3944  ∅c0 4321  𝒫 cpw 4601  {csn 4627  dom cdm 5675  Fun wfun 6536  ‘cfv 6542  Vtxcvtx 28523  iEdgciedg 28524  UHGraphcuhgr 28583

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-uhgr 28585

This theorem is referenced by:  lpvtx  28595  upgrle2  28632  uhgredgiedgb  28653  uhgriedg0edg0  28654  uhgrvtxedgiedgb  28663  edglnl  28670  numedglnl  28671  uhgr2edg  28732  ushgredgedg  28753  ushgredgedgloop  28755  0uhgrsubgr  28803  uhgrsubgrself  28804  subgruhgrfun  28806  subgruhgredgd  28808  subumgredg2  28809  subupgr  28811  uhgrspansubgrlem  28814  uhgrspansubgr  28815  uhgrspan1  28827  upgrreslem  28828  umgrreslem  28829  upgrres  28830  umgrres  28831  vtxduhgr0e  29002  vtxduhgrun  29007  vtxduhgrfiun  29008  finsumvtxdg2ssteplem1  29069  upgrewlkle2  29130  upgredginwlk  29160  wlkiswwlks1  29388  wlkiswwlksupgr2  29398  umgrwwlks2on  29478  vdn0conngrumgrv2  29716  eulerpathpr  29760  eulercrct  29762  lfuhgr  34406  loop1cycl  34426  umgr2cycllem  34429