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Theorem upgrres1lem1 27102
Description: Lemma 1 for upgrres1 27106. (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 6675 . . 3 𝑉 ∈ V
32difexi 5218 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
5 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
65fvexi 6675 . . . 4 𝐸 ∈ V
74, 6rabex2 5223 . . 3 𝐹 ∈ V
8 resiexg 7614 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
97, 8ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
103, 9pm3.2i 474 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  wnel 3118  {crab 3137  Vcvv 3480  cdif 3916  {csn 4550   I cid 5446  cres 5544  cfv 6343  Vtxcvtx 26792  Edgcedg 26843
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-res 5554  df-iota 6302  df-fv 6351
This theorem is referenced by:  upgrres1lem2  27104  upgrres1lem3  27105
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