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Theorem uhgrspan1lem1 28554
Description: Lemma 1 for uhgrspan1 28557. (Contributed by AV, 19-Nov-2020.)
Hypotheses
Ref Expression
uhgrspan1.v 𝑉 = (Vtx‘𝐺)
uhgrspan1.i 𝐼 = (iEdg‘𝐺)
uhgrspan1.f 𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}
Assertion
Ref Expression
uhgrspan1lem1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)

Proof of Theorem uhgrspan1lem1
StepHypRef Expression
1 uhgrspan1.v . . . 4 𝑉 = (Vtx‘𝐺)
21fvexi 6905 . . 3 𝑉 ∈ V
32difexi 5328 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
54fvexi 6905 . . 3 𝐼 ∈ V
65resex 6029 . 2 (𝐼𝐹) ∈ V
73, 6pm3.2i 471 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  wnel 3046  {crab 3432  Vcvv 3474  cdif 3945  {csn 4628  dom cdm 5676  cres 5678  cfv 6543  Vtxcvtx 28253  iEdgciedg 28254
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-ext 2703  ax-sep 5299  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-res 5688  df-iota 6495  df-fv 6551
This theorem is referenced by:  uhgrspan1lem2  28555  uhgrspan1lem3  28556
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