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Theorem uhgriedg0edg0 26016
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 2621 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 25955 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 eqid 2621 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
41, 3edg0iedg0 25943 . 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 1482  wcel 1989  c0 3913  Fun wfun 5880  cfv 5886  iEdgciedg 25869  Edgcedg 25933   UHGraph cuhgr 25945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-edg 25934  df-uhgr 25947
This theorem is referenced by:  uhgr0v0e  26124  uhgr0vusgr  26128  lfuhgr1v0e  26140  usgr1vr  26141  usgr1v0e  26212  uhgr0edg0rgr  26463  rgrusgrprc  26479
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