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Theorem uhgriedg0edg0 28927
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 2727 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 28866 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 eqid 2727 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
41, 3edg0iedg0 28855 . 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 205   = wceq 1534  wcel 2099  c0 4318  Fun wfun 6536  cfv 6542  iEdgciedg 28797  Edgcedg 28847  UHGraphcuhgr 28856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-edg 28848  df-uhgr 28858
This theorem is referenced by:  uhgr0v0e  29038  uhgr0vusgr  29042  lfuhgr1v0e  29054  usgr1vr  29055  usgr1v0e  29126  uhgr0edg0rgr  29374  rgrusgrprc  29390
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