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Theorem upgrres1lem1 26089
Description: Lemma 1 for upgrres1 26093. (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 6158 . . . 4 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2694 . . 3 𝑉 ∈ V
43difexi 4769 . 2 (𝑉 ∖ {𝑁}) ∈ V
5 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
6 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
7 fvex 6158 . . . . 5 (Edg‘𝐺) ∈ V
86, 7eqeltri 2694 . . . 4 𝐸 ∈ V
95, 8rabex2 4775 . . 3 𝐹 ∈ V
10 resiexg 7049 . . 3 (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
119, 10ax-mp 5 . 2 ( I ↾ 𝐹) ∈ V
124, 11pm3.2i 471 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  wnel 2893  {crab 2911  Vcvv 3186  cdif 3552  {csn 4148   I cid 4984  cres 5076  cfv 5847  Vtxcvtx 25774  Edgcedg 25839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-res 5086  df-iota 5810  df-fv 5855
This theorem is referenced by:  upgrres1lem2  26091  upgrres1lem3  26092
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