Theorem uspgrupgr 29221

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 2737	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2737	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29195	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		f1f 6812	. . . 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 29127	. . 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 2108  {crab 3436   ∖ cdif 3963  ∅c0 4342  𝒫 cpw 4608  {csn 4634   class class class wbr 5151  dom cdm 5693  ⟶wf 6565  –1-1→wf1 6566  ‘cfv 6569   ≤ cle 11303  2c2 12328  ♯chash 14375  Vtxcvtx 29039  iEdgciedg 29040  UPGraphcupgr 29123  USPGraphcuspgr 29191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fv 6577  df-upgr 29125  df-uspgr 29193

This theorem is referenced by:  uspgrupgrushgr  29222  uspgruhgr  29227  usgrupgr  29228  uspgrun  29231  uspgrunop  29232  uspgredg2vtxeu  29263  1loopgrnb0  29546  uspgr2wlkeq  29690  uspgrn2crct  29854  wlkiswwlks2  29921  wlkiswwlks  29922  wlklnwwlkn  29930  clwlkclwwlk  30047  wlk2v2e  30202  isuspgrim0  47838  isuspgrimlem  47840  uspgropssxp  48026  uspgrsprf  48028