Theorem upgrres1lem3 26426

Description: Lemma 3 for upgrres1 26427. (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 6335	. 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 26423	. . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
7		opiedgfv 26107	. . 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 2792	1 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)

Syntax hints:   ∧ wa 382   = wceq 1630   ∈ wcel 2144   ∉ wnel 3045  {crab 3064  Vcvv 3349   ∖ cdif 3718  {csn 4314  ⟨cop 4320   I cid 5156   ↾ cres 5251  ‘cfv 6031  Vtxcvtx 26094  iEdgciedg 26095  Edgcedg 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-2nd 7315  df-iedg 26097

This theorem is referenced by:  upgrres1  26427  umgrres1  26428  usgrres1  26429  nbupgrres  26487