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Theorem edg0usgr 29270
Description: A class without edges is a simple graph. Since ran 𝐹 = ∅ does not generally imply Fun 𝐹, but Fun (iEdg‘𝐺) is required for 𝐺 to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)
Assertion
Ref Expression
edg0usgr ((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)

Proof of Theorem edg0usgr
StepHypRef Expression
1 edgval 29066 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . 4 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
32eqeq1d 2739 . . 3 (𝐺𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4 funrel 6583 . . . . . 6 (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5 relrn0 5983 . . . . . . 7 (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
65bicomd 223 . . . . . 6 (Rel (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
74, 6syl 17 . . . . 5 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
8 simpr 484 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺𝑊)
9 simpl 482 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → (iEdg‘𝐺) = ∅)
108, 9usgr0e 29253 . . . . . 6 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺 ∈ USGraph)
1110ex 412 . . . . 5 ((iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph))
127, 11biimtrdi 253 . . . 4 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph)))
1312com13 88 . . 3 (𝐺𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
143, 13sylbid 240 . 2 (𝐺𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15143imp 1111 1 ((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  c0 4333  ran crn 5686  Rel wrel 5690  Fun wfun 6555  cfv 6561  iEdgciedg 29014  Edgcedg 29064  USGraphcusgr 29166
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-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  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-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-edg 29065  df-usgr 29168
This theorem is referenced by: (None)
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