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Theorem edg0usgr 26550
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 26347 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . 4 (𝐺𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
32eqeq1d 2827 . . 3 (𝐺𝑊 → ((Edg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
4 funrel 6140 . . . . . 6 (Fun (iEdg‘𝐺) → Rel (iEdg‘𝐺))
5 relrn0 5616 . . . . . . 7 (Rel (iEdg‘𝐺) → ((iEdg‘𝐺) = ∅ ↔ ran (iEdg‘𝐺) = ∅))
65bicomd 215 . . . . . 6 (Rel (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
74, 6syl 17 . . . . 5 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
8 simpr 479 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺𝑊)
9 simpl 476 . . . . . . 7 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → (iEdg‘𝐺) = ∅)
108, 9usgr0e 26533 . . . . . 6 (((iEdg‘𝐺) = ∅ ∧ 𝐺𝑊) → 𝐺 ∈ USGraph)
1110ex 403 . . . . 5 ((iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph))
127, 11syl6bi 245 . . . 4 (Fun (iEdg‘𝐺) → (ran (iEdg‘𝐺) = ∅ → (𝐺𝑊𝐺 ∈ USGraph)))
1312com13 88 . . 3 (𝐺𝑊 → (ran (iEdg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
143, 13sylbid 232 . 2 (𝐺𝑊 → ((Edg‘𝐺) = ∅ → (Fun (iEdg‘𝐺) → 𝐺 ∈ USGraph)))
15143imp 1141 1 ((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  c0 4144  ran crn 5343  Rel wrel 5347  Fun wfun 6117  cfv 6123  iEdgciedg 26295  Edgcedg 26345  USGraphcusgr 26448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fv 6131  df-edg 26346  df-usgr 26450
This theorem is referenced by: (None)
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