Theorem upgruhgr 29395

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 2769	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2769	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 29379	. . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		ssrab2 4042	. . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		fss 6725	. . 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 597	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	1, 2	isuhgr 29353	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	6, 7	mpbird 260	1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

Syntax hints:   → wi 4   ∈ wcel 2149  {crab 3423   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594   class class class wbr 5113  dom cdm 5664  ⟶wf 6535  ‘cfv 6539   ≤ cle 11246  2c2 12297  ♯chash 14368  Vtxcvtx 29289  iEdgciedg 29290  UHGraphcuhgr 29349  UPGraphcupgr 29373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5273

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-uhgr 29351  df-upgr 29375

This theorem is referenced by:  umgruhgr  29397  upgrle2  29398  edglnl  29436  numedglnl  29437  uspgruhgr  29477  usgruhgr  29479  subupgr  29580  upgrspan  29586  upgrreslem  29597  upgrres  29599  finsumvtxdg2ssteplem1  29838  finsumvtxdg2size  29843  upgrewlkle2  29899  upgredginwlk  29928  wlkiswwlks1  30159  wlkiswwlksupgr2  30169  eulerpathpr  30534  eulercrct  30536  upgracycumgr  35580  isubgrupgr  48561