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Theorem upgrres1lem1 40528
 Description: Lemma 1 for upgrres1 40532. (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)
2 fvex 6113 . . . 4 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2684 . . 3 𝑉 ∈ V
43difexi 4736 . 2 (𝑉 ∖ {𝑁}) ∈ V
5 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
6 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
7 fvex 6113 . . . . 5 (Edg‘𝐺) ∈ V
86, 7eqeltri 2684 . . . 4 𝐸 ∈ V
95, 8rabex2 4742 . . 3 𝐹 ∈ V
10 resiexg 6994 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
119, 10ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
124, 11pm3.2i 470 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125   I cid 4948   ↾ cres 5040  ‘cfv 5804  Vtxcvtx 25673  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050  df-iota 5768  df-fv 5812 This theorem is referenced by:  upgrres1lem2  40530  upgrres1lem3  40531
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