Theorem upgruhgr 29191

Description: An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)

Assertion

Ref	Expression
upgruhgr	⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

Proof of Theorem upgruhgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2737	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 29175	. . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		ssrab2 4034	. . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		fss 6686	. . 3 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	3, 4, 5	sylancl 587	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	1, 2	isuhgr 29149	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	6, 7	mpbird 257	1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

Syntax hints:   → wi 4   ∈ wcel 2114  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582   class class class wbr 5100  dom cdm 5632  ⟶wf 6496  ‘cfv 6500   ≤ cle 11179  2c2 12212  ♯chash 14265  Vtxcvtx 29085  iEdgciedg 29086  UHGraphcuhgr 29145  UPGraphcupgr 29169

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 2709  ax-nul 5253

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-uhgr 29147  df-upgr 29171

This theorem is referenced by:  umgruhgr  29193  upgrle2  29194  edglnl  29232  numedglnl  29233  uspgruhgr  29273  usgruhgr  29275  subupgr  29376  upgrspan  29382  upgrreslem  29393  upgrres  29395  finsumvtxdg2ssteplem1  29635  finsumvtxdg2size  29640  upgrewlkle2  29696  upgredginwlk  29725  wlkiswwlks1  29956  wlkiswwlksupgr2  29966  eulerpathpr  30331  eulercrct  30333  upgracycumgr  35373  isubgrupgr  48234