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Theorem upgrres1lem1 29326
Description: Lemma 1 for upgrres1 29330. (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 6920 . . 3 𝑉 ∈ V
32difexi 5330 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
5 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
65fvexi 6920 . . . 4 𝐸 ∈ V
74, 6rabex2 5341 . . 3 𝐹 ∈ V
8 resiexg 7934 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
97, 8ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
103, 9pm3.2i 470 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  wnel 3046  {crab 3436  Vcvv 3480  cdif 3948  {csn 4626   I cid 5577  cres 5687  cfv 6561  Vtxcvtx 29013  Edgcedg 29064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697  df-iota 6514  df-fv 6569
This theorem is referenced by:  upgrres1lem2  29328  upgrres1lem3  29329
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