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

Theorem uspgrupgr 28474
Description: A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
Assertion
Ref Expression
uspgrupgr (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)

Proof of Theorem uspgrupgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2732 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isuspgr 28450 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4 f1f 6787 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
53, 4syl6bi 252 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
61, 2isupgr 28382 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
75, 6sylibrd 258 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph))
87pm2.43i 52 1 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  {crab 3432  cdif 3945  c0 4322  𝒫 cpw 4602  {csn 4628   class class class wbr 5148  dom cdm 5676  wf 6539  1-1wf1 6540  cfv 6543  cle 11251  2c2 12269  chash 14292  Vtxcvtx 28294  iEdgciedg 28295  UPGraphcupgr 28378  USPGraphcuspgr 28446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551  df-upgr 28380  df-uspgr 28448
This theorem is referenced by:  uspgrupgrushgr  28475  usgrupgr  28480  uspgrun  28483  uspgrunop  28484  uspgredg2vtxeu  28515  1loopgrnb0  28797  uspgr2wlkeq  28941  uspgrn2crct  29100  wlkiswwlks2  29167  wlkiswwlks  29168  wlklnwwlkn  29176  clwlkclwwlk  29293  wlk2v2e  29448  isomuspgrlem1  46574  isomuspgrlem2b  46576  isomuspgrlem2c  46577  isomuspgrlem2d  46578  uspgropssxp  46601  uspgrsprf  46603
  Copyright terms: Public domain W3C validator