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Theorem usgruspgr 28171
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 2733 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2733 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2isusgr 28146 . . . 4 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
4 2re 12232 . . . . . . . 8 2 ∈ ℝ
54eqlei2 11271 . . . . . . 7 ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)
65a1i 11 . . . . . 6 (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2))
76ss2rabi 4035 . . . . 5 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
8 f1ss 6745 . . . . 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 690 . . . 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 253 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
111, 2isuspgr 28145 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1210, 11sylibrd 259 . 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 1542  wcel 2107  {crab 3406  cdif 3908  wss 3911  c0 4283  𝒫 cpw 4561  {csn 4587   class class class wbr 5106  dom cdm 5634  1-1wf1 6494  cfv 6497  cle 11195  2c2 12213  chash 14236  Vtxcvtx 27989  iEdgciedg 27990  USPGraphcuspgr 28141  USGraphcusgr 28142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-i2m1 11124  ax-1ne0 11125  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-2 12221  df-uspgr 28143  df-usgr 28144
This theorem is referenced by:  usgrumgruspgr  28173  usgruspgrb  28174  usgrupgr  28175  usgrislfuspgr  28177  usgredg2vtxeu  28211  usgredgedg  28220  usgredgleord  28223  vtxdusgrfvedg  28481  usgrn2cycl  28796  wlksnfi  28894  rusgrnumwwlk  28962  rusgrnumwlkg  28964  clwlksndivn  29072  clwlknon2num  29354  numclwlk1lem2  29356
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