Theorem uhgrfun 29149

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 29145	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6666	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887  ∅c0 4274  𝒫 cpw 4542  {csn 4568  dom cdm 5624  Fun wfun 6486  ‘cfv 6492  Vtxcvtx 29079  iEdgciedg 29080  UHGraphcuhgr 29139

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 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-uhgr 29141

This theorem is referenced by:  lpvtx  29151  upgrle2  29188  uhgredgiedgb  29209  uhgriedg0edg0  29210  uhgrvtxedgiedgb  29219  edglnl  29226  numedglnl  29227  uhgr2edg  29291  ushgredgedg  29312  ushgredgedgloop  29314  0uhgrsubgr  29362  uhgrsubgrself  29363  subgruhgrfun  29365  subgruhgredgd  29367  subumgredg2  29368  subupgr  29370  uhgrspansubgrlem  29373  uhgrspansubgr  29374  uhgrspan1  29386  upgrreslem  29387  umgrreslem  29388  upgrres  29389  umgrres  29390  vtxduhgr0e  29562  vtxduhgrun  29567  vtxduhgrfiun  29568  finsumvtxdg2ssteplem1  29629  upgrewlkle2  29690  upgredginwlk  29719  wlkiswwlks1  29950  wlkiswwlksupgr2  29960  usgrwwlks2on  30041  umgrwwlks2on  30042  vdn0conngrumgrv2  30281  eulerpathpr  30325  eulercrct  30327  lfuhgr  35316  loop1cycl  35335  umgr2cycllem  35338  isubgrvtxuhgr  48352  isubgredg  48354  isubgrsubgr  48357  isubgr0uhgr  48361  uhgrimedgi  48378  isuspgrim0lem  48381  isuspgrim0  48382  upgrimwlklem2  48386  upgrimwlklem3  48387  upgrimtrlslem1  48392  clnbgrgrimlem  48421  clnbgrgrim  48422  grimedg  48423