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Theorem uhgrspan1lem1 28297
Description: Lemma 1 for uhgrspan1 28300. (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 6860 . . 3 𝑉 ∈ V
32difexi 5289 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
54fvexi 6860 . . 3 𝐼 ∈ V
65resex 5989 . 2 (𝐼𝐹) ∈ V
73, 6pm3.2i 472 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  wnel 3046  {crab 3406  Vcvv 3447  cdif 3911  {csn 4590  dom cdm 5637  cres 5639  cfv 6500  Vtxcvtx 27996  iEdgciedg 27997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-sn 4591  df-pr 4593  df-uni 4870  df-res 5649  df-iota 6452  df-fv 6508
This theorem is referenced by:  uhgrspan1lem2  28298  uhgrspan1lem3  28299
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