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Theorem uhgrspanop 28542
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 5463 . . 3 𝑉, (𝐸𝐴)⟩ ∈ V
43a1i 11 . 2 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ V)
51fvexi 6902 . . . 4 𝑉 ∈ V
62fvexi 6902 . . . . 5 𝐸 ∈ V
76resex 6027 . . . 4 (𝐸𝐴) ∈ V
85, 7opvtxfvi 28258 . . 3 (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉
98a1i 11 . 2 (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉)
105, 7opiedgfvi 28259 . . 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 28538 1 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3474  cop 4633  cres 5677  cfv 6540  Vtxcvtx 28245  iEdgciedg 28246  UHGraphcuhgr 28305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-1st 7971  df-2nd 7972  df-vtx 28247  df-iedg 28248  df-edg 28297  df-uhgr 28307  df-subgr 28514
This theorem is referenced by: (None)
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