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Theorem uhgriedg0edg0 26432
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 2825 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 26371 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 eqid 2825 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
41, 3edg0iedg0 26360 . 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 198   = wceq 1656  wcel 2164  c0 4146  Fun wfun 6121  cfv 6127  iEdgciedg 26302  Edgcedg 26352  UHGraphcuhgr 26361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-edg 26353  df-uhgr 26363
This theorem is referenced by:  uhgr0v0e  26542  uhgr0vusgr  26546  lfuhgr1v0e  26558  usgr1vr  26559  usgr1v0e  26630  uhgr0edg0rgr  26878  rgrusgrprc  26894
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