Theorem ushgruhgr 29142

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 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	ushgrf 29136	. . 3 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		f1f 6730	. . 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 29133	. 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 2113   ∖ cdif 3898  ∅c0 4285  𝒫 cpw 4554  {csn 4580  dom cdm 5624  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  Vtxcvtx 29069  iEdgciedg 29070  UHGraphcuhgr 29129  USHGraphcushgr 29130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fv 6500  df-uhgr 29131  df-ushgr 29132

This theorem is referenced by:  ushgrun  29149  ushgrunop  29150  ushgredgedg  29302  ushgredgedgloop  29304  ushggricedg  48183