Theorem uhgrfun 25961

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 e. UHGraph -> Fun E )

Proof of Theorem uhgrfun

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2		uhgrfun.e	. . 3 |- E = (iEdg ` G )
3	1, 2	uhgrf 25957	. 2 |- ( G e. UHGraph -> E : dom E --> ( ~P (Vtx ` G ) \ { (/) } ) )
4	3	ffund 6049	1 |- ( G e. UHGraph -> Fun E )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   Fun wfun 5882   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-uhgr 25953

This theorem is referenced by:  lpvtx  25963  upgrle2  26000  uhgredgiedgb  26021  uhgriedg0edg0  26022  uhgrvtxedgiedgb  26031  edglnl  26038  numedglnl  26039  uhgr2edg  26100  ushgredgedg  26121  ushgredgedgloop  26123  0uhgrsubgr  26171  uhgrsubgrself  26172  subgruhgrfun  26174  subgruhgredgd  26176  subumgredg2  26177  subupgr  26179  uhgrspansubgrlem  26182  uhgrspansubgr  26183  uhgrspan1  26195  upgrreslem  26196  umgrreslem  26197  upgrres  26198  umgrres  26199  vtxduhgr0e  26374  vtxduhgrun  26379  vtxduhgrfiun  26380  finsumvtxdg2ssteplem1  26441  upgrewlkle2  26502  upgredginwlk  26532  wlkiswwlks1  26753  wlkiswwlksupgr2  26763  umgrwwlks2on  26850  vdn0conngrumgrv2  27056  eulerpathpr  27100  eulercrct  27102