Theorem upgruhgr 29680

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 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2760	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 29664	. . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		ssrab2 4005	. . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		fss 6726	. . 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 598	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	1, 2	isuhgr 29638	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	6, 7	mpbird 260	1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  {crab 3412   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4583   class class class wbr 5102  dom cdm 5650  ⟶wf 6534  ‘cfv 6538   ≤ cle 11344  2c2 12397  ♯chash 14474  Vtxcvtx 29574  iEdgciedg 29575  UHGraphcuhgr 29634  UPGraphcupgr 29658

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 4584  df-pr 4586  df-op 4590  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 6540  df-fn 6541  df-f 6542  df-fv 6546  df-uhgr 29636  df-upgr 29660

This theorem is used by:  umgruhgr  29682  upgrle2  29683  edglnl  29721  numedglnl  29722  uspgruhgr  29765  usgruhgr  29767  subupgr  29868  upgrspan  29874  upgrreslem  29885  upgrres  29887  finsumvtxdg2ssteplem1  30126  finsumvtxdg2size  30131  upgrewlkle2  30187  upgredginwlk  30216  wlkiswwlks1  30456  wlkiswwlksupgr2  30466  eulerpathpr  30841  eulercrct  30843  upgracycumgr  35918  isubgrupgr  48968