Theorem ushgruhgr 29476

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 2765	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2765	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	ushgrf 29470	. . 3 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		f1f 6778	. . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
5	3, 4	syl 18	. 2 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	1, 2	isuhgr 29467	. 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
7	5, 6	mpbird 260	1 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ∖ cdif 3903  ∅c0 4286  𝒫 cpw 4564  {csn 4591  dom cdm 5663  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  Vtxcvtx 29403  iEdgciedg 29404  UHGraphcuhgr 29463  USHGraphcushgr 29464

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548  df-uhgr 29465  df-ushgr 29466

This theorem is used by:  ushgrun  29483  ushgrunop  29484  ushgredgedg  29639  ushgredgedgloop  29641  ushggricedg  48752