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Theorem upgrres1lem1 29161
Description: Lemma 1 for upgrres1 29165. (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 6904 . . 3 𝑉 ∈ V
32difexi 5326 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
5 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
65fvexi 6904 . . . 4 𝐸 ∈ V
74, 6rabex2 5332 . . 3 𝐹 ∈ V
8 resiexg 7914 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
97, 8ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
103, 9pm3.2i 469 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  wnel 3036  {crab 3419  Vcvv 3463  cdif 3938  {csn 4625   I cid 5570  cres 5675  cfv 6543  Vtxcvtx 28848  Edgcedg 28899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-res 5685  df-iota 6495  df-fv 6551
This theorem is referenced by:  upgrres1lem2  29163  upgrres1lem3  29164
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