Theorem ushgruhgr 29154

Description: An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Assertion

Ref	Expression
ushgruhgr	⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)

Proof of Theorem ushgruhgr

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2737	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	ushgrf 29148	. . 3 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		f1f 6738	. . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
5	3, 4	syl 17	. 2 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	1, 2	isuhgr 29145	. 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
7	5, 6	mpbird 257	1 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)

Syntax hints:   → wi 4   ∈ wcel 2114   ∖ cdif 3900  ∅c0 4287  𝒫 cpw 4556  {csn 4582  dom cdm 5632  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500  Vtxcvtx 29081  iEdgciedg 29082  UHGraphcuhgr 29141  USHGraphcushgr 29142

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-f1 6505  df-fv 6508  df-uhgr 29143  df-ushgr 29144

This theorem is referenced by:  ushgrun  29161  ushgrunop  29162  ushgredgedg  29314  ushgredgedgloop  29316  ushggricedg  48287