Theorem uspgrupgr 29765

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 2760	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2760	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29739	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		f1f 6779	. . . 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 256	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
6	1, 2	isupgr 29668	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
7	5, 6	sylibrd 262	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph))
8	7	pm2.43i 53	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  {crab 3412   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4584   class class class wbr 5102  dom cdm 5650  ⟶wf 6535  –1-1→wf1 6536  ‘cfv 6539   ≤ cle 11348  2c2 12401  ♯chash 14478  Vtxcvtx 29580  iEdgciedg 29581  UPGraphcupgr 29664  USPGraphcuspgr 29735

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fv 6547  df-upgr 29666  df-uspgr 29737

This theorem is used by:  uspgrupgrushgr  29766  uspgruhgr  29771  usgrupgr  29772  uspgrun  29775  uspgrunop  29776  uspgredg2vtxeu  29807  1loopgrnb0  30089  uspgr2wlkeq  30232  uspgrn2crct  30403  wlkiswwlks2  30470  wlkiswwlks  30471  wlklnwwlkn  30479  clwlkclwwlk  30599  wlk2v2e  30764  isuspgrim0  49086  isuspgrimlem  49087  upgrimwlklem5  49093  upgrimwlk  49094  grlimprclnbgr  49188  grlimprclnbgrvtx  49191  grlimgredgex  49192  uspgropssxp  49336  uspgrsprf  49338