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Theorem edg0iedg0 29090
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 29084 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
31, 2eqtri 2768 . . . 4 𝐸 = ran (iEdg‘𝐺)
43eqeq1i 2745 . . 3 (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)
54a1i 11 . 2 (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2749 . . . . 5 (iEdg‘𝐺) = 𝐼
87rneqi 5962 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
98eqeq1i 2745 . . 3 (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)
109a1i 11 . 2 (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 6595 . . 3 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5995 . . . 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 1537  c0 4352  ran crn 5701  Rel wrel 5705  Fun wfun 6567  cfv 6573  iEdgciedg 29032  Edgcedg 29082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-edg 29083
This theorem is referenced by:  uhgriedg0edg0  29162  egrsubgr  29312  vtxduhgr0e  29514
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