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Theorem upgrres1lem1 28566
Description: Lemma 1 for upgrres1 28570. (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 6906 . . 3 𝑉 ∈ V
32difexi 5329 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
5 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
65fvexi 6906 . . . 4 𝐸 ∈ V
74, 6rabex2 5335 . . 3 𝐹 ∈ V
8 resiexg 7905 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
97, 8ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
103, 9pm3.2i 472 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  wnel 3047  {crab 3433  Vcvv 3475  cdif 3946  {csn 4629   I cid 5574  cres 5679  cfv 6544  Vtxcvtx 28256  Edgcedg 28307
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-res 5689  df-iota 6496  df-fv 6552
This theorem is referenced by:  upgrres1lem2  28568  upgrres1lem3  28569
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