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Theorem upgrres1lem3 29344
Description: Lemma 3 for upgrres1 29345. (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
upgrres1lem3 (iEdg‘𝑆) = ( I ↾ 𝐹)
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉
Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem upgrres1lem3
StepHypRef Expression
1 upgrres1.s . . 3 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
21fveq2i 6910 . 2 (iEdg‘𝑆) = (iEdg‘⟨(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 29341 . . 3 ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
7 opiedgfv 29039 . . 3 (((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) → (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = ( I ↾ 𝐹))
86, 7ax-mp 5 . 2 (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = ( I ↾ 𝐹)
92, 8eqtri 2763 1 (iEdg‘𝑆) = ( I ↾ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  wnel 3044  {crab 3433  Vcvv 3478  cdif 3960  {csn 4631  cop 4637   I cid 5582  cres 5691  cfv 6563  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-2nd 8014  df-iedg 29031
This theorem is referenced by:  upgrres1  29345  umgrres1  29346  usgrres1  29347  nbupgrres  29396
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