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Theorem uspgrupgr 28700
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 2731 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2731 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isuspgr 28676 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4 f1f 6788 . . . 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 28608 . . 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 2105  {crab 3431  cdif 3946  c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  dom cdm 5677  wf 6540  1-1wf1 6541  cfv 6544  cle 11254  2c2 12272  chash 14295  Vtxcvtx 28520  iEdgciedg 28521  UPGraphcupgr 28604  USPGraphcuspgr 28672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552  df-upgr 28606  df-uspgr 28674
This theorem is referenced by:  uspgrupgrushgr  28701  usgrupgr  28706  uspgrun  28709  uspgrunop  28710  uspgredg2vtxeu  28741  1loopgrnb0  29023  uspgr2wlkeq  29167  uspgrn2crct  29326  wlkiswwlks2  29393  wlkiswwlks  29394  wlklnwwlkn  29402  clwlkclwwlk  29519  wlk2v2e  29674  isomuspgrlem1  46795  isomuspgrlem2b  46797  isomuspgrlem2c  46798  isomuspgrlem2d  46799  uspgropssxp  46822  uspgrsprf  46824
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