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Theorem uhgrspanop 27198
 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 5328 . . 3 𝑉, (𝐸𝐴)⟩ ∈ V
43a1i 11 . 2 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ V)
51fvexi 6677 . . . 4 𝑉 ∈ V
62fvexi 6677 . . . . 5 𝐸 ∈ V
76resex 5876 . . . 4 (𝐸𝐴) ∈ V
85, 7opvtxfvi 26914 . . 3 (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉
98a1i 11 . 2 (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸𝐴)⟩) = 𝑉)
105, 7opiedgfvi 26915 . . 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 27194 1 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   ↾ cres 5530  ‘cfv 6340  Vtxcvtx 26901  iEdgciedg 26902  UHGraphcuhgr 26961 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-1st 7699  df-2nd 7700  df-vtx 26903  df-iedg 26904  df-edg 26953  df-uhgr 26963  df-subgr 27170 This theorem is referenced by: (None)
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