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Theorem edg0iedg0 29460
Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.)
Hypotheses
Ref Expression
edg0iedg0.i 𝐼 = (iEdg‘𝐺)
edg0iedg0.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
edg0iedg0 (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))

Proof of Theorem edg0iedg0
StepHypRef Expression
1 edg0iedg0.e . . . . 5 𝐸 = (Edg‘𝐺)
2 edgval 29454 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2788 . . . 4 𝐸 = ran (iEdg‘𝐺)
43eqeq1i 2770 . . 3 (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)
54a1i 11 . 2 (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2774 . . . . 5 (iEdg‘𝐺) = 𝐼
87rneqi 5929 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
98eqeq1i 2770 . . 3 (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
109a1i 11 . 2 (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 6557 . . 3 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5965 . . . 4 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 226 . . 3 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 18 . 2 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
155, 10, 143bitrd 308 1 (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wb 209   = wceq 1570  c0 4286  ran crn 5664  Rel wrel 5668  Fun wfun 6534  cfv 6540  iEdgciedg 29402  Edgcedg 29452
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-edg 29453
This theorem is used by:  uhgriedg0edg0  29532  egrsubgr  29685  vtxduhgr0e  29886
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