Theorem isuhgrop 29535

Description: The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Assertion

Ref	Expression
isuhgrop	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isuhgrop

Step	Hyp	Ref	Expression
1		opex 5443	. . 3 ⊢ ⟨𝑉, 𝐸⟩ ∈ V
2		eqid 2762	. . . 4 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3		eqid 2762	. . . 4 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
4	2, 3	isuhgr 29525	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ V → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)⟶(𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅})))
5	1, 4	mp1i 14	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)⟶(𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅})))
6		opiedgfv 29472	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
7	6	dmeqd 5893	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8		opvtxfv 29469	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
9	8	pweqd 4577	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
10	9	difeq1d 4076	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
11	6, 7, 10	feq123d 6695	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ((iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)⟶(𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
12	5, 11	bitrd 282	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899  ∅c0 4282  𝒫 cpw 4560  {csn 4587  ⟨cop 4593  dom cdm 5659  ⟶wf 6533  ‘cfv 6537  Vtxcvtx 29461  iEdgciedg 29462  UHGraphcuhgr 29521

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

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-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-1st 7990  df-2nd 7991  df-vtx 29463  df-iedg 29464  df-uhgr 29523

This theorem is used by:  pliguhgr  30975