Theorem upgrres1lem3 29893

Description: Lemma 3 for upgrres1 29894. (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 6888	. 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 29890	. . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
7		opiedgfv 29585	. . 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 2783	1 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061  {crab 3412  Vcvv 3450   ∖ cdif 4045  {csn 4583  ⟨cop 4589   I cid 5544   ↾ cres 5652  ‘cfv 6538  Vtxcvtx 29574  iEdgciedg 29575  Edgcedg 29625

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-2nd 8002  df-iedg 29577

This theorem is used by:  upgrres1  29894  umgrres1  29895  usgrres1  29896  nbupgrres  29945