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Theorem upgrres1lem2 27104
 Description: Lemma 2 for upgrres1 27106. (Contributed by AV, 7-Nov-2020.)
Hypotheses
Ref Expression
upgrres1.v 𝑉 = (Vtx‘𝐺)
upgrres1.e 𝐸 = (Edg‘𝐺)
upgrres1.f 𝐹 = {𝑒𝐸𝑁𝑒}
upgrres1.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
Assertion
Ref Expression
upgrres1lem2 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉
Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem upgrres1lem2
StepHypRef Expression
1 upgrres1.s . . 3 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
21fveq2i 6652 . 2 (Vtx‘𝑆) = (Vtx‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩)
3 upgrres1.v . . . 4 𝑉 = (Vtx‘𝐺)
4 upgrres1.e . . . 4 𝐸 = (Edg‘𝐺)
5 upgrres1.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
63, 4, 5upgrres1lem1 27102 . . 3 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
7 opvtxfv 26800 . . 3 (((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) → (Vtx‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = (𝑉 ∖ {𝑁}))
86, 7ax-mp 5 . 2 (Vtx‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = (𝑉 ∖ {𝑁})
92, 8eqtri 2824 1 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ∉ wnel 3094  {crab 3113  Vcvv 3444   ∖ cdif 3881  {csn 4528  ⟨cop 4534   I cid 5427   ↾ cres 5525  ‘cfv 6328  Vtxcvtx 26792  Edgcedg 26843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7675  df-vtx 26794 This theorem is referenced by:  upgrres1  27106  umgrres1  27107  usgrres1  27108  nbupgrres  27157  nbupgruvtxres  27200  uvtxupgrres  27201  cusgrres  27241
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