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Theorem upgrres1lem1 28555
Description: Lemma 1 for upgrres1 28559. (Contributed by AV, 7-Nov-2020.)
Hypotheses
Ref Expression
upgrres1.v 𝑉 = (Vtx‘𝐺)
upgrres1.e 𝐸 = (Edg‘𝐺)
upgrres1.f 𝐹 = {𝑒𝐸𝑁𝑒}
Assertion
Ref Expression
upgrres1lem1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉
Allowed substitution hint:   𝐹(𝑒)

Proof of Theorem upgrres1lem1
StepHypRef Expression
1 upgrres1.v . . . 4 𝑉 = (Vtx‘𝐺)
21fvexi 6902 . . 3 𝑉 ∈ V
32difexi 5327 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
5 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
65fvexi 6902 . . . 4 𝐸 ∈ V
74, 6rabex2 5333 . . 3 𝐹 ∈ V
8 resiexg 7901 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
97, 8ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
103, 9pm3.2i 471 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  wnel 3046  {crab 3432  Vcvv 3474  cdif 3944  {csn 4627   I cid 5572  cres 5677  cfv 6540  Vtxcvtx 28245  Edgcedg 28296
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 5298  ax-nul 5305  ax-pow 5362  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-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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-id 5573  df-xp 5681  df-rel 5682  df-res 5687  df-iota 6492  df-fv 6548
This theorem is referenced by:  upgrres1lem2  28557  upgrres1lem3  28558
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