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Theorem umgr0e 27201
Description: The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
Hypotheses
Ref Expression
umgr0e.g (𝜑𝐺𝑊)
umgr0e.e (𝜑 → (iEdg‘𝐺) = ∅)
Assertion
Ref Expression
umgr0e (𝜑𝐺 ∈ UMGraph)

Proof of Theorem umgr0e
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 umgr0e.e . . . 4 (𝜑 → (iEdg‘𝐺) = ∅)
21f10d 6694 . . 3 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 f1f 6615 . . 3 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
42, 3syl 17 . 2 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
5 umgr0e.g . . 3 (𝜑𝐺𝑊)
6 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isumgr 27186 . . 3 (𝐺𝑊 → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
95, 8syl 17 . 2 (𝜑 → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
104, 9mpbird 260 1 (𝜑𝐺 ∈ UMGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  {crab 3065  cdif 3863  c0 4237  𝒫 cpw 4513  {csn 4541  dom cdm 5551  wf 6376  1-1wf1 6377  cfv 6380  2c2 11885  chash 13896  Vtxcvtx 27087  iEdgciedg 27088  UMGraphcumgr 27172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fv 6388  df-umgr 27174
This theorem is referenced by:  upgr0e  27202
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