Theorem uhgrfun 26006

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 2651	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 26002	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	ffund 6087	1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ∖ cdif 3604  ∅c0 3948  𝒫 cpw 4191  {csn 4210  dom cdm 5143  Fun wfun 5920  ‘cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  UHGraphcuhgr 25996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-uhgr 25998

This theorem is referenced by:  lpvtx  26008  upgrle2  26045  uhgredgiedgb  26066  uhgriedg0edg0  26067  uhgrvtxedgiedgb  26076  edglnl  26083  numedglnl  26084  uhgr2edg  26145  ushgredgedg  26166  ushgredgedgloop  26168  0uhgrsubgr  26216  uhgrsubgrself  26217  subgruhgrfun  26219  subgruhgredgd  26221  subumgredg2  26222  subupgr  26224  uhgrspansubgrlem  26227  uhgrspansubgr  26228  uhgrspan1  26240  upgrreslem  26241  umgrreslem  26242  upgrres  26243  umgrres  26244  vtxduhgr0e  26430  vtxduhgrun  26435  vtxduhgrfiun  26436  finsumvtxdg2ssteplem1  26497  upgrewlkle2  26558  upgredginwlk  26588  wlkiswwlks1  26821  wlkiswwlksupgr2  26831  umgrwwlks2on  26923  vdn0conngrumgrv2  27174  eulerpathpr  27218  eulercrct  27220