Theorem uspgrupgr 26071

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 2622	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2622	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 26047	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
4		f1f 6101	. . . 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	syl6bi 243	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
6	1, 2	isupgr 25979	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
7	5, 6	sylibrd 249	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph ))
8	7	pm2.43i 52	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )

Syntax hints:   → wi 4   ∈ wcel 1990  {crab 2916   ∖ cdif 3571  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888   ≤ cle 10075  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975   USPGraph cuspgr 26043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-upgr 25977  df-uspgr 26045

This theorem is referenced by:  uspgrupgrushgr  26072  usgrupgr  26077  uspgrun  26080  uspgrunop  26081  uspgredg2vtxeu  26112  1loopgrnb0  26398  uspgr2wlkeq  26542  uspgrn2crct  26700  wlkiswwlks2  26761  wlkiswwlks  26762  wlklnwwlkn  26770  wlknwwlksninj  26775  wlknwwlksnsur  26776  wlkwwlkinj  26782  wlkwwlksur  26783  clwlkclwwlk  26903  wlk2v2e  27017  uspgropssxp  41752  uspgrsprf  41754