Theorem uspgrupgr 29433

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 2765	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2765	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29407	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		f1f 6764	. . . 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 29339	. . 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)

Syntax hints:   → wi 4   ∈ wcel 2145  {crab 3417   ∖ cdif 3904  ∅c0 4288  𝒫 cpw 4558  {csn 4585   class class class wbr 5104  dom cdm 5651  ⟶wf 6521  –1-1→wf1 6522  ‘cfv 6525   ≤ cle 11232  2c2 12283  ♯chash 14354  Vtxcvtx 29251  iEdgciedg 29252  UPGraphcupgr 29335  USPGraphcuspgr 29403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fv 6533  df-upgr 29337  df-uspgr 29405

This theorem is referenced by:  uspgrupgrushgr  29434  uspgruhgr  29439  usgrupgr  29440  uspgrun  29443  uspgrunop  29444  uspgredg2vtxeu  29475  1loopgrnb0  29757  uspgr2wlkeq  29900  uspgrn2crct  30062  wlkiswwlks2  30129  wlkiswwlks  30130  wlklnwwlkn  30138  clwlkclwwlk  30258  wlk2v2e  30413  isuspgrim0  48515  isuspgrimlem  48516  upgrimwlklem5  48522  upgrimwlk  48523  grlimprclnbgr  48617  grlimprclnbgrvtx  48620  grlimgredgex  48621  uspgropssxp  48765  uspgrsprf  48767