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Theorem uhgrspanop 29328
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 5475 . . 3 𝑉, (𝐸𝐴)⟩ ∈ V
43a1i 11 . 2 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ V)
51fvexi 6921 . . . 4 𝑉 ∈ V
62fvexi 6921 . . . . 5 𝐸 ∈ V
76resex 6049 . . . 4 (𝐸𝐴) ∈ V
85, 7opvtxfvi 29041 . . 3 (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉
98a1i 11 . 2 (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉)
105, 7opiedgfvi 29042 . . 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 29324 1 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cres 5691  cfv 6563  Vtxcvtx 29028  iEdgciedg 29029  UHGraphcuhgr 29088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1st 8013  df-2nd 8014  df-vtx 29030  df-iedg 29031  df-edg 29080  df-uhgr 29090  df-subgr 29300
This theorem is referenced by: (None)
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