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Theorem uhgrspanop 29313
Description: A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
Hypotheses
Ref Expression
uhgrspanop.v 𝑉 = (Vtx‘𝐺)
uhgrspanop.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
uhgrspanop (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)

Proof of Theorem uhgrspanop
StepHypRef Expression
1 uhgrspanop.v . 2 𝑉 = (Vtx‘𝐺)
2 uhgrspanop.e . 2 𝐸 = (iEdg‘𝐺)
3 opex 5469 . . 3 𝑉, (𝐸𝐴)⟩ ∈ V
43a1i 11 . 2 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ V)
51fvexi 6920 . . . 4 𝑉 ∈ V
62fvexi 6920 . . . . 5 𝐸 ∈ V
76resex 6047 . . . 4 (𝐸𝐴) ∈ V
85, 7opvtxfvi 29026 . . 3 (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉
98a1i 11 . 2 (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉)
105, 7opiedgfvi 29027 . . 3 (iEdg‘⟨𝑉, (𝐸𝐴)⟩) = (𝐸𝐴)
1110a1i 11 . 2 (𝐺 ∈ UHGraph → (iEdg‘⟨𝑉, (𝐸𝐴)⟩) = (𝐸𝐴))
12 id 22 . 2 (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
131, 2, 4, 9, 11, 12uhgrspan 29309 1 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cres 5687  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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  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-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-subgr 29285
This theorem is referenced by: (None)
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