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Theorem uhgruhgra 40375
 Description: Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
Assertion
Ref Expression
uhgruhgra ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝑉 UHGrph 𝐸)

Proof of Theorem uhgruhgra
StepHypRef Expression
1 eqid 2610 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2610 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2uhgrf 25728 . . . 4 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
433ad2ant1 1075 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
5 simp3 1056 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝐸 = (iEdg‘𝐺))
65dmeqd 5248 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → dom 𝐸 = dom (iEdg‘𝐺))
7 pweq 4111 . . . . . 6 (𝑉 = (Vtx‘𝐺) → 𝒫 𝑉 = 𝒫 (Vtx‘𝐺))
873ad2ant2 1076 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝒫 𝑉 = 𝒫 (Vtx‘𝐺))
98difeq1d 3689 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐺) ∖ {∅}))
105, 6, 9feq123d 5947 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
114, 10mpbird 246 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
12 fvex 6113 . . . . . 6 (Vtx‘𝐺) ∈ V
13 eleq1 2676 . . . . . 6 (𝑉 = (Vtx‘𝐺) → (𝑉 ∈ V ↔ (Vtx‘𝐺) ∈ V))
1412, 13mpbiri 247 . . . . 5 (𝑉 = (Vtx‘𝐺) → 𝑉 ∈ V)
15 fvex 6113 . . . . . 6 (iEdg‘𝐺) ∈ V
16 eleq1 2676 . . . . . 6 (𝐸 = (iEdg‘𝐺) → (𝐸 ∈ V ↔ (iEdg‘𝐺) ∈ V))
1715, 16mpbiri 247 . . . . 5 (𝐸 = (iEdg‘𝐺) → 𝐸 ∈ V)
1814, 17anim12i 588 . . . 4 ((𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
19183adant1 1072 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
20 isuhgra 25827 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
2119, 20syl 17 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
2211, 21mpbird 246 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝑉 UHGrph 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   UHGrph cuhg 25819 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-mo 2463  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  df-uhgra 25821 This theorem is referenced by:  uhgrauhgrbi  40377
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