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Theorem edg0iedg0 25800
 Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.)
Hypotheses
Ref Expression
edg0iedg0.i 𝐼 = (iEdg‘𝐺)
edg0iedg0.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
edg0iedg0 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

Proof of Theorem edg0iedg0
StepHypRef Expression
1 edg0iedg0.e . . . . 5 𝐸 = (Edg‘𝐺)
2 edgaval 25794 . . . . 5 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2656 . . . 4 (𝐺𝑊𝐸 = ran (iEdg‘𝐺))
43adantr 480 . . 3 ((𝐺𝑊 ∧ Fun 𝐼) → 𝐸 = ran (iEdg‘𝐺))
54eqeq1d 2612 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅))
6 edg0iedg0.i . . . . . 6 𝐼 = (iEdg‘𝐺)
76eqcomi 2619 . . . . 5 (iEdg‘𝐺) = 𝐼
87a1i 11 . . . 4 ((𝐺𝑊 ∧ Fun 𝐼) → (iEdg‘𝐺) = 𝐼)
98rneqd 5274 . . 3 ((𝐺𝑊 ∧ Fun 𝐼) → ran (iEdg‘𝐺) = ran 𝐼)
109eqeq1d 2612 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅))
11 funrel 5821 . . . 4 (Fun 𝐼 → Rel 𝐼)
12 relrn0 5304 . . . . 5 (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅))
1312bicomd 212 . . . 4 (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1411, 13syl 17 . . 3 (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
1514adantl 481 . 2 ((𝐺𝑊 ∧ Fun 𝐼) → (ran 𝐼 = ∅ ↔ 𝐼 = ∅))
165, 10, 153bitrd 293 1 ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ran crn 5039  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804  iEdgciedg 25674  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-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-fv 5812  df-edga 25793 This theorem is referenced by:  uhgriedg0edg0  25801  egrsubgr  40501  vtxduhgr0e  40693
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