Theorem upgrres1lem2 29661

Description: Lemma 2 for upgrres1 29663. (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
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

Step	Hyp	Ref	Expression
1		upgrres1.s	. . 3 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
2	1	fveq2i 6884	. 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 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6	3, 4, 5	upgrres1lem1 29659	. . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
7		opvtxfv 29354	. . 3 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) → (Vtx‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = (𝑉 ∖ {𝑁}))
8	6, 7	ax-mp 5	. 2 ⊢ (Vtx‘⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩) = (𝑉 ∖ {𝑁})
9	2, 8	eqtri 2786	1 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∉ wnel 3064  {crab 3416  Vcvv 3455   ∖ cdif 3902  {csn 4589  ⟨cop 4595   I cid 5555   ↾ cres 5663  ‘cfv 6536  Vtxcvtx 29346  Edgcedg 29397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-1st 7982  df-vtx 29348

This theorem is referenced by:  upgrres1  29663  umgrres1  29664  usgrres1  29665  nbupgrres  29714  nbupgruvtxres  29757  uvtxupgrres  29758  cusgrres  29798