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Theorem edg0usgr 26190
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 25986 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . 4 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
32eqeq1d 2653 . . 3 (𝐺𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4 funrel 5943 . . . . . 6 (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5 relrn0 5415 . . . . . . 7 (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
65bicomd 213 . . . . . 6 (Rel (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
74, 6syl 17 . . . . 5 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
8 simpr 476 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺𝑊)
9 simpl 472 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → (iEdg‘𝐺) = ∅)
108, 9usgr0e 26173 . . . . . 6 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺 ∈ USGraph)
1110ex 449 . . . . 5 ((iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph))
127, 11syl6bi 243 . . . 4 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph)))
1312com13 88 . . 3 (𝐺𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
143, 13sylbid 230 . 2 (𝐺𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15143imp 1275 1 ((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  c0 3948  ran crn 5144  Rel wrel 5148  Fun wfun 5920  cfv 5926  iEdgciedg 25920  Edgcedg 25984  USGraphcusgr 26089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934  df-edg 25985  df-usgr 26091
This theorem is referenced by: (None)
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