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

Theorem umgr0e 25999
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 6168 . . 3 (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
3 f1f 6099 . . 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 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isumgr 25984 . . 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 247 1 (𝜑𝐺 ∈ UMGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  {crab 2915  cdif 3569  c0 3913  𝒫 cpw 4156  {csn 4175  dom cdm 5112  wf 5882  1-1wf1 5883  cfv 5886  2c2 11067  #chash 13112  Vtxcvtx 25868  iEdgciedg 25869   UMGraph cumgr 25970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fv 5894  df-umgr 25972
This theorem is referenced by:  upgr0e  26000
  Copyright terms: Public domain W3C validator