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Theorem uspgrupgr 28433
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 28409 . . . 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 28341 . . 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 11248  2c2 12266  chash 14289  Vtxcvtx 28253  iEdgciedg 28254  UPGraphcupgr 28337  USPGraphcuspgr 28405
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 28339  df-uspgr 28407
This theorem is referenced by:  uspgrupgrushgr  28434  usgrupgr  28439  uspgrun  28442  uspgrunop  28443  uspgredg2vtxeu  28474  1loopgrnb0  28756  uspgr2wlkeq  28900  uspgrn2crct  29059  wlkiswwlks2  29126  wlkiswwlks  29127  wlklnwwlkn  29135  clwlkclwwlk  29252  wlk2v2e  29407  isomuspgrlem1  46485  isomuspgrlem2b  46487  isomuspgrlem2c  46488  isomuspgrlem2d  46489  uspgropssxp  46512  uspgrsprf  46514
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