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Theorem usgruspgr 26272
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 2760 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2760 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isusgr 26247 . . . 4 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
4 2re 11282 . . . . . . . 8 2 ∈ ℝ
54eqlei2 10340 . . . . . . 7 ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)
65a1i 11 . . . . . 6 (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2))
76ss2rabi 3825 . . . . 5 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
8 f1ss 6267 . . . . 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 709 . . . 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, 9syl6bi 243 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
111, 2isuspgr 26246 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1210, 11sylibrd 249 . 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 1632  wcel 2139  {crab 3054  cdif 3712  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   class class class wbr 4804  dom cdm 5266  1-1wf1 6046  cfv 6049  cle 10267  2c2 11262  chash 13311  Vtxcvtx 26073  iEdgciedg 26074  USPGraphcuspgr 26242  USGraphcusgr 26243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-i2m1 10196  ax-1ne0 10197  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-2 11271  df-uspgr 26244  df-usgr 26245
This theorem is referenced by:  usgrumgruspgr  26274  usgruspgrb  26275  usgrupgr  26276  usgrislfuspgr  26278  usgredg2vtxeu  26312  usgredgedg  26321  usgredgleord  26324  vtxdusgrfvedg  26597  usgrn2cycl  26912  wlksnfi  27025  wlksnwwlknvbij  27026  rusgrnumwwlk  27097  clwlksfoclwwlkOLD  27217  clwlksf1clwwlkOLD  27223  clwlksndivn  27231  clwlknon2num  27529  numclwlk1lem2  27531
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