Theorem uhgreq12g 27338

Description: If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)

Hypotheses

Ref	Expression
uhgrf.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrf.e	⊢ 𝐸 = (iEdg‘𝐺)
uhgreq12g.w	⊢ 𝑊 = (Vtx‘𝐻)
uhgreq12g.f	⊢ 𝐹 = (iEdg‘𝐻)

Assertion

Ref	Expression
uhgreq12g	⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))

Proof of Theorem uhgreq12g

Step	Hyp	Ref	Expression
1		uhgrf.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrf.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isuhgr 27333	. . . 4 ⊢ (𝐺 ∈ 𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4	3	adantr 480	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
5	4	adantr 480	. 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
6		simpr 484	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐸 = 𝐹)
7	6	dmeqd 5803	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8		pweq 4546	. . . . . 6 ⊢ (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
9	8	difeq1d 4052	. . . . 5 ⊢ (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
10	9	adantr 480	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
11	6, 7, 10	feq123d 6573	. . 3 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
12		uhgreq12g.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐻)
13		uhgreq12g.f	. . . . . 6 ⊢ 𝐹 = (iEdg‘𝐻)
14	12, 13	isuhgr 27333	. . . . 5 ⊢ (𝐻 ∈ 𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
15	14	adantl 481	. . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
16	15	bicomd 222	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
17	11, 16	sylan9bbr 510	. 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
18	5, 17	bitrd 278	1 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880  ∅c0 4253  𝒫 cpw 4530  {csn 4558  dom cdm 5580  ⟶wf 6414  ‘cfv 6418  Vtxcvtx 27269  iEdgciedg 27270  UHGraphcuhgr 27329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-uhgr 27331

This theorem is referenced by: (None)