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Theorem edg0usgr 29018
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 28817 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . 4 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
32eqeq1d 2728 . . 3 (𝐺𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4 funrel 6559 . . . . . 6 (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5 relrn0 5962 . . . . . . 7 (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
65bicomd 222 . . . . . 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 29001 . . . . . 6 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺 ∈ USGraph)
1110ex 412 . . . . 5 ((iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph))
127, 11biimtrdi 252 . . . 4 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph)))
1312com13 88 . . 3 (𝐺𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
143, 13sylbid 239 . 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  c0 4317  ran crn 5670  Rel wrel 5674  Fun wfun 6531  cfv 6537  iEdgciedg 28765  Edgcedg 28815  USGraphcusgr 28917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fv 6545  df-edg 28816  df-usgr 28919
This theorem is referenced by: (None)
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