MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  umgr0e Structured version   Visualization version   GIF version

Theorem umgr0e 28995
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 6872 . . 3 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3 f1f 6793 . . 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 2725 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2725 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isumgr 28980 . . 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 256 1 (𝜑𝐺 ∈ UMGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {crab 3418  cdif 3941  c0 4322  𝒫 cpw 4604  {csn 4630  dom cdm 5678  wf 6545  1-1wf1 6546  cfv 6549  2c2 12300  chash 14325  Vtxcvtx 28881  iEdgciedg 28882  UMGraphcumgr 28966
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fv 6557  df-umgr 28968
This theorem is referenced by:  upgr0e  28996
  Copyright terms: Public domain W3C validator