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Theorem uspgrupgr 27835
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 2736 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2736 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isuspgr 27811 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4 f1f 6721 . . . 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 27743 . . 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 3403  cdif 3895  c0 4269  𝒫 cpw 4547  {csn 4573   class class class wbr 5092  dom cdm 5620  wf 6475  1-1wf1 6476  cfv 6479  cle 11111  2c2 12129  chash 14145  Vtxcvtx 27655  iEdgciedg 27656  UPGraphcupgr 27739  USPGraphcuspgr 27807
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 2707  ax-nul 5250
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fv 6487  df-upgr 27741  df-uspgr 27809
This theorem is referenced by:  uspgrupgrushgr  27836  usgrupgr  27841  uspgrun  27844  uspgrunop  27845  uspgredg2vtxeu  27876  1loopgrnb0  28158  uspgr2wlkeq  28302  uspgrn2crct  28461  wlkiswwlks2  28528  wlkiswwlks  28529  wlklnwwlkn  28537  clwlkclwwlk  28654  wlk2v2e  28809  isomuspgrlem1  45638  isomuspgrlem2b  45640  isomuspgrlem2c  45641  isomuspgrlem2d  45642  uspgropssxp  45665  uspgrsprf  45667
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