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Theorem usgruspgr 29037
Description: A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
Assertion
Ref Expression
usgruspgr (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)

Proof of Theorem usgruspgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2725 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isusgr 29010 . . . 4 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
4 2re 12316 . . . . . . . 8 2 ∈ ℝ
54eqlei2 11355 . . . . . . 7 ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)
65a1i 11 . . . . . 6 (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2))
76ss2rabi 4066 . . . . 5 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
8 f1ss 6794 . . . . 5 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
97, 8mpan2 689 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
103, 9biimtrdi 252 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
111, 2isuspgr 29009 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1210, 11sylibrd 258 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph))
1312pm2.43i 52 1 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3419  cdif 3936  wss 3939  c0 4318  𝒫 cpw 4598  {csn 4624   class class class wbr 5143  dom cdm 5672  1-1wf1 6540  cfv 6543  cle 11279  2c2 12297  chash 14321  Vtxcvtx 28853  iEdgciedg 28854  USPGraphcuspgr 29005  USGraphcusgr 29006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-i2m1 11206  ax-1ne0 11207  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-2 12305  df-uspgr 29007  df-usgr 29008
This theorem is referenced by:  usgrumgruspgr  29039  usgruspgrb  29040  usgrupgr  29042  usgrislfuspgr  29044  usgredg2vtxeu  29078  usgredgedg  29087  usgredgleord  29090  vtxdusgrfvedg  29349  usgrn2cycl  29664  wlksnfi  29762  rusgrnumwwlk  29830  rusgrnumwlkg  29832  clwlksndivn  29940  clwlknon2num  30222  numclwlk1lem2  30224
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