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Theorem edg0iedg0 29072
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 29066 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2765 . . . 4 𝐸 = ran (iEdg‘𝐺)
43eqeq1i 2742 . . 3 (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)
54a1i 11 . 2 (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2746 . . . . 5 (iEdg‘𝐺) = 𝐼
87rneqi 5948 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
98eqeq1i 2742 . . 3 (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
109a1i 11 . 2 (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 6583 . . 3 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5983 . . . 4 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 223 . . 3 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 17 . 2 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
155, 10, 143bitrd 305 1 (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  c0 4333  ran crn 5686  Rel wrel 5690  Fun wfun 6555  cfv 6561  iEdgciedg 29014  Edgcedg 29064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-edg 29065
This theorem is referenced by:  uhgriedg0edg0  29144  egrsubgr  29294  vtxduhgr0e  29496
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