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Theorem edg0usgr 27041
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 26840 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . 4 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
32eqeq1d 2824 . . 3 (𝐺𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4 funrel 6351 . . . . . 6 (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5 relrn0 5818 . . . . . . 7 (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
65bicomd 226 . . . . . 6 (Rel (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
74, 6syl 17 . . . . 5 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
8 simpr 488 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺𝑊)
9 simpl 486 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → (iEdg‘𝐺) = ∅)
108, 9usgr0e 27024 . . . . . 6 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺 ∈ USGraph)
1110ex 416 . . . . 5 ((iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph))
127, 11syl6bi 256 . . . 4 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph)))
1312com13 88 . . 3 (𝐺𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
143, 13sylbid 243 . 2 (𝐺𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15143imp 1108 1 ((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  c0 4265  ran crn 5533  Rel wrel 5537  Fun wfun 6328  cfv 6334  iEdgciedg 26788  Edgcedg 26838  USGraphcusgr 26940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fv 6342  df-edg 26839  df-usgr 26942
This theorem is referenced by: (None)
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