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Theorem uspgrupgr 25998
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 2621 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2621 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isuspgr 25974 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
4 f1f 6068 . . . 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 243 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
61, 2isupgr 25909 . . 3 (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
75, 6sylibrd 249 . 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 1987  {crab 2912  cdif 3557  c0 3897  𝒫 cpw 4136  {csn 4155   class class class wbr 4623  dom cdm 5084  wf 5853  1-1wf1 5854  cfv 5857  cle 10035  2c2 11030  #chash 13073  Vtxcvtx 25808  iEdgciedg 25809   UPGraph cupgr 25905   USPGraph cuspgr 25970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fv 5865  df-upgr 25907  df-uspgr 25972
This theorem is referenced by:  uspgrupgrushgr  25999  usgrupgr  26004  uspgrun  26007  uspgrunop  26008  uspgredg2vtxeu  26039  1loopgrnb0  26318  uspgr2wlkeq  26445  uspgrn2crct  26603  wlkiswwlks2  26664  wlkiswwlks  26665  wlklnwwlkn  26673  wlknwwlksninj  26678  wlknwwlksnsur  26679  wlkwwlkinj  26685  wlkwwlksur  26686  clwlkclwwlk  26804  wlk2v2e  26917  uspgropssxp  41070  uspgrsprf  41072
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