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Theorem edg0iedg0 28180
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 28174 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2759 . . . 4 𝐸 = ran (iEdg‘𝐺)
43eqeq1i 2736 . . 3 (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)
54a1i 11 . 2 (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2740 . . . . 5 (iEdg‘𝐺) = 𝐼
87rneqi 5928 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
98eqeq1i 2736 . . 3 (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
109a1i 11 . 2 (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 6554 . . 3 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5960 . . . 4 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 222 . . 3 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 17 . 2 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
155, 10, 143bitrd 304 1 (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  c0 4318  ran crn 5670  Rel wrel 5674  Fun wfun 6526  cfv 6532  iEdgciedg 28122  Edgcedg 28172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fv 6540  df-edg 28173
This theorem is referenced by:  uhgriedg0edg0  28252  egrsubgr  28399  vtxduhgr0e  28600
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