Theorem uhgriedg0edg0 25801

Description: A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 15-Dec-2020.)

Assertion

Ref	Expression
uhgriedg0edg0	⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Proof of Theorem uhgriedg0edg0

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	uhgrfun 25732	. 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3		eqid 2610	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	1, 3	edg0iedg0 25800	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺)) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
5	2, 4	mpdan 699	1 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∅c0 3874  Fun wfun 5798  ‘cfv 5804  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724  df-edga 25793

This theorem is referenced by:  uhgr0v0e  40464  uhgr0vusgr  40468  lfuhgr1v0e  40480  usgr1vr  40481  usgr1v0e  40545  uhgr0edg0rgr  40773  rgrusgrprc  40789