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Theorem umgr0e 29128
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 6881 . . 3 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 f1f 6803 . . 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 2736 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2736 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isumgr 29113 . . 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 257 1 (𝜑𝐺 ∈ UMGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  {crab 3435  cdif 3947  c0 4332  𝒫 cpw 4599  {csn 4625  dom cdm 5684  wf 6556  1-1wf1 6557  cfv 6560  2c2 12322  chash 14370  Vtxcvtx 29014  iEdgciedg 29015  UMGraphcumgr 29099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fv 6568  df-umgr 29101
This theorem is referenced by:  upgr0e  29129
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