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Theorem upgrres1lem1 26371
Description: Lemma 1 for upgrres1 26375. (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 6350 . . . 4 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2823 . . 3 𝑉 ∈ V
43difexi 4949 . 2 (𝑉 ∖ {𝑁}) ∈ V
5 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
6 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
7 fvex 6350 . . . . 5 (Edg‘𝐺) ∈ V
86, 7eqeltri 2823 . . . 4 𝐸 ∈ V
95, 8rabex2 4954 . . 3 𝐹 ∈ V
10 resiexg 7255 . . 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 1620  wcel 2127  wnel 3023  {crab 3042  Vcvv 3328  cdif 3700  {csn 4309   I cid 5161  cres 5256  cfv 6037  Vtxcvtx 26044  Edgcedg 26109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-res 5266  df-iota 6000  df-fv 6045
This theorem is referenced by:  upgrres1lem2  26373  upgrres1lem3  26374
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