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Theorem uhgrspan1lem1 27009
Description: Lemma 1 for uhgrspan1 27012. (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 6677 . . 3 𝑉 ∈ V
32difexi 5223 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
54fvexi 6677 . . 3 𝐼 ∈ V
65resex 5892 . 2 (𝐼𝐹) ∈ V
73, 6pm3.2i 471 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  wnel 3120  {crab 3139  Vcvv 3492  cdif 3930  {csn 4557  dom cdm 5548  cres 5550  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-uni 4831  df-res 5560  df-iota 6307  df-fv 6356
This theorem is referenced by:  uhgrspan1lem2  27010  uhgrspan1lem3  27011
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