Theorem uhgreq12g 27424

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 27419	. . . 4 ⊢ (𝐺 ∈ 𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4	3	adantr 481	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
5	4	adantr 481	. 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
6		simpr 485	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐸 = 𝐹)
7	6	dmeqd 5809	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8		pweq 4551	. . . . . 6 ⊢ (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
9	8	difeq1d 4057	. . . . 5 ⊢ (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
10	9	adantr 481	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
11	6, 7, 10	feq123d 6583	. . 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 27419	. . . . 5 ⊢ (𝐻 ∈ 𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
15	14	adantl 482	. . . 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 511	. 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 396   = wceq 1539   ∈ wcel 2106   ∖ cdif 3885  ∅c0 4258  𝒫 cpw 4535  {csn 4563  dom cdm 5586  ⟶wf 6424  ‘cfv 6428  Vtxcvtx 27355  iEdgciedg 27356  UHGraphcuhgr 27415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-opab 5138  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-fv 6436  df-uhgr 27417

This theorem is referenced by: (None)