Theorem uhgrfun 29135

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 29131	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6672	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567  dom cdm 5631  Fun wfun 6492  ‘cfv 6498  Vtxcvtx 29065  iEdgciedg 29066  UHGraphcuhgr 29125

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 2708  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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-uhgr 29127

This theorem is referenced by:  lpvtx  29137  upgrle2  29174  uhgredgiedgb  29195  uhgriedg0edg0  29196  uhgrvtxedgiedgb  29205  edglnl  29212  numedglnl  29213  uhgr2edg  29277  ushgredgedg  29298  ushgredgedgloop  29300  0uhgrsubgr  29348  uhgrsubgrself  29349  subgruhgrfun  29351  subgruhgredgd  29353  subumgredg2  29354  subupgr  29356  uhgrspansubgrlem  29359  uhgrspansubgr  29360  uhgrspan1  29372  upgrreslem  29373  umgrreslem  29374  upgrres  29375  umgrres  29376  vtxduhgr0e  29547  vtxduhgrun  29552  vtxduhgrfiun  29553  finsumvtxdg2ssteplem1  29614  upgrewlkle2  29675  upgredginwlk  29704  wlkiswwlks1  29935  wlkiswwlksupgr2  29945  usgrwwlks2on  30026  umgrwwlks2on  30027  vdn0conngrumgrv2  30266  eulerpathpr  30310  eulercrct  30312  lfuhgr  35300  loop1cycl  35319  umgr2cycllem  35322  isubgrvtxuhgr  48340  isubgredg  48342  isubgrsubgr  48345  isubgr0uhgr  48349  uhgrimedgi  48366  isuspgrim0lem  48369  isuspgrim0  48370  upgrimwlklem2  48374  upgrimwlklem3  48375  upgrimtrlslem1  48380  clnbgrgrimlem  48409  clnbgrgrim  48410  grimedg  48411