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Theorem uhgreq12g 29082
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 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))

Proof of Theorem uhgreq12g
StepHypRef Expression
1 uhgrf.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 uhgrf.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2isuhgr 29077 . . . 4 (𝐺𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
43adantr 480 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
54adantr 480 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
6 simpr 484 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → 𝐸 = 𝐹)
76dmeqd 5916 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8 pweq 4614 . . . . . 6 (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
98difeq1d 4125 . . . . 5 (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
109adantr 480 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
116, 7, 10feq123d 6725 . . 3 ((𝑉 = 𝑊𝐸 = 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)
1412, 13isuhgr 29077 . . . . 5 (𝐻𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1514adantl 481 . . . 4 ((𝐺𝑋𝐻𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1615bicomd 223 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
1711, 16sylan9bbr 510 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
185, 17bitrd 279 1 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  wf 6557  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-uhgr 29075
This theorem is referenced by: (None)
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