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Theorem uhgriedg0edg0 26067
Description: A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.)
Assertion
Ref Expression
uhgriedg0edg0 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Proof of Theorem uhgriedg0edg0
StepHypRef Expression
1 eqid 2651 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 26006 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 eqid 2651 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
41, 3edg0iedg0 25994 . 2 (Fun (iEdg‘𝐺) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
52, 4syl 17 1 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  c0 3948  Fun wfun 5920  cfv 5926  iEdgciedg 25920  Edgcedg 25984  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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
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-mo 2503  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-csb 3567  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-mpt 4763  df-id 5053  df-xp 5149  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-edg 25985  df-uhgr 25998
This theorem is referenced by:  uhgr0v0e  26175  uhgr0vusgr  26179  lfuhgr1v0e  26191  usgr1vr  26192  usgr1v0e  26263  uhgr0edg0rgr  26525  rgrusgrprc  26541
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