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Theorem uhgreq12g 25731
 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 25726 . . . 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 476 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → 𝐸 = 𝐹)
76dmeqd 5248 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8 pweq 4111 . . . . . 6 (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
98difeq1d 3689 . . . . 5 (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
109adantr 480 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
116, 7, 10feq123d 5947 . . 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 25726 . . . . 5 (𝐻𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1514adantl 481 . . . 4 ((𝐺𝑋𝐻𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1615bicomd 212 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph ))
1711, 16sylan9bbr 733 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph ))
185, 17bitrd 267 1 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724 This theorem is referenced by: (None)
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