Theorem uspgrupgr 29247

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 2736	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29221	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		f1f 6736	. . . 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 29153	. . 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 2114  {crab 3389   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567   class class class wbr 5085  dom cdm 5631  ⟶wf 6494  –1-1→wf1 6495  ‘cfv 6498   ≤ cle 11180  2c2 12236  ♯chash 14292  Vtxcvtx 29065  iEdgciedg 29066  UPGraphcupgr 29149  USPGraphcuspgr 29217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fv 6506  df-upgr 29151  df-uspgr 29219

This theorem is referenced by:  uspgrupgrushgr  29248  uspgruhgr  29253  usgrupgr  29254  uspgrun  29257  uspgrunop  29258  uspgredg2vtxeu  29289  1loopgrnb0  29571  uspgr2wlkeq  29714  uspgrn2crct  29876  wlkiswwlks2  29943  wlkiswwlks  29944  wlklnwwlkn  29952  clwlkclwwlk  30072  wlk2v2e  30227  isuspgrim0  48370  isuspgrimlem  48371  upgrimwlklem5  48377  upgrimwlk  48378  grlimprclnbgr  48472  grlimprclnbgrvtx  48475  grlimgredgex  48476  uspgropssxp  48620  uspgrsprf  48622