Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrfun Structured version   Visualization version   GIF version

Theorem uhgrfun 25857
 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
StepHypRef Expression
1 eqid 2621 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 uhgrfun.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 25853 . 2 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
43ffund 6006 1 (𝐺 ∈ UHGraph → Fun 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987   ∖ cdif 3552  ∅c0 3891  𝒫 cpw 4130  {csn 4148  dom cdm 5074  Fun wfun 5841  ‘cfv 5847  Vtxcvtx 25774  iEdgciedg 25775   UHGraph cuhgr 25847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-uhgr 25849 This theorem is referenced by:  lpvtx  25859  upgrle2  25895  uhgredgiedgb  25916  uhgriedg0edg0  25917  uhgrvtxedgiedgb  25926  uhgr2edg  25993  ushgredgedg  26014  ushgredgedgloop  26016  0uhgrsubgr  26064  uhgrsubgrself  26065  subgruhgrfun  26067  subgruhgredgd  26069  subumgredg2  26070  subupgr  26072  uhgrspansubgrlem  26075  uhgrspansubgr  26076  uhgrspan1  26088  vtxduhgr0e  26260  vtxduhgrun  26265  vtxduhgrfiun  26266  upgrewlkle2  26372  upgredginwlk  26401  wlkiswwlks1  26622  wlkiswwlksupgr2  26632  umgrwwlks2on  26719  vdn0conngrumgrv2  26922  eulerpathpr  26966  eulercrct  26968
 Copyright terms: Public domain W3C validator