Theorem uspgrupgr 29114

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 2726	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2726	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29088	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		f1f 6798	. . . 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 252	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
6	1, 2	isupgr 29020	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
7	5, 6	sylibrd 258	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph))
8	7	pm2.43i 52	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)

Syntax hints:   → wi 4   ∈ wcel 2099  {crab 3419   ∖ cdif 3944  ∅c0 4325  𝒫 cpw 4607  {csn 4633   class class class wbr 5153  dom cdm 5682  ⟶wf 6550  –1-1→wf1 6551  ‘cfv 6554   ≤ cle 11299  2c2 12319  ♯chash 14347  Vtxcvtx 28932  iEdgciedg 28933  UPGraphcupgr 29016  USPGraphcuspgr 29084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fv 6562  df-upgr 29018  df-uspgr 29086

This theorem is referenced by:  uspgrupgrushgr  29115  uspgruhgr  29120  usgrupgr  29121  uspgrun  29124  uspgrunop  29125  uspgredg2vtxeu  29156  1loopgrnb0  29439  uspgr2wlkeq  29583  uspgrn2crct  29742  wlkiswwlks2  29809  wlkiswwlks  29810  wlklnwwlkn  29818  clwlkclwwlk  29935  wlk2v2e  30090  isuspgrim0  47451  isuspgrimlem  47453  uspgropssxp  47521  uspgrsprf  47523