Theorem upgrres1lem3 27660

Description: Lemma 3 for upgrres1 27661. (Contributed by AV, 7-Nov-2020.)

Hypotheses

Ref	Expression
upgrres1.v	⊢ 𝑉 = (Vtx‘𝐺)
upgrres1.e	⊢ 𝐸 = (Edg‘𝐺)
upgrres1.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
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

Step	Hyp	Ref	Expression
1		upgrres1.s	. . 3 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
2	1	fveq2i 6771	. 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 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6	3, 4, 5	upgrres1lem1 27657	. . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
7		opiedgfv 27358	. . 3 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) → (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = ( I ↾ 𝐹))
8	6, 7	ax-mp 5	. 2 ⊢ (iEdg‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = ( I ↾ 𝐹)
9	2, 8	eqtri 2767	1 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2109   ∉ wnel 3050  {crab 3069  Vcvv 3430   ∖ cdif 3888  {csn 4566  ⟨cop 4572   I cid 5487   ↾ cres 5590  ‘cfv 6430  Vtxcvtx 27347  iEdgciedg 27348  Edgcedg 27398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fun 6432  df-fv 6438  df-2nd 7818  df-iedg 27350

This theorem is referenced by:  upgrres1  27661  umgrres1  27662  usgrres1  27663  nbupgrres  27712