Theorem uspgrupgr 29160

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.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29134	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		f1f 6726	. . . 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})
5	3, 4	biimtrdi 253	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
6	1, 2	isupgr 29066	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
7	5, 6	sylibrd 259	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph))
8	7	pm2.43i 52	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)

Syntax hints:   → wi 4   ∈ wcel 2113  {crab 3396   ∖ cdif 3895  ∅c0 4282  𝒫 cpw 4551  {csn 4577   class class class wbr 5095  dom cdm 5621  ⟶wf 6484  –1-1→wf1 6485  ‘cfv 6488   ≤ cle 11156  2c2 12189  ♯chash 14241  Vtxcvtx 28978  iEdgciedg 28979  UPGraphcupgr 29062  USPGraphcuspgr 29130

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fv 6496  df-upgr 29064  df-uspgr 29132

This theorem is referenced by:  uspgrupgrushgr  29161  uspgruhgr  29166  usgrupgr  29167  uspgrun  29170  uspgrunop  29171  uspgredg2vtxeu  29202  1loopgrnb0  29485  uspgr2wlkeq  29628  uspgrn2crct  29790  wlkiswwlks2  29857  wlkiswwlks  29858  wlklnwwlkn  29866  clwlkclwwlk  29986  wlk2v2e  30141  isuspgrim0  48021  isuspgrimlem  48022  upgrimwlklem5  48028  upgrimwlk  48029  grlimprclnbgr  48123  grlimprclnbgrvtx  48126  grlimgredgex  48127  uspgropssxp  48271  uspgrsprf  48273