Theorem stgrfv 47855

Description: The star graph S_N. (Contributed by AV, 10-Sep-2025.)

Assertion

Ref	Expression
stgrfv	⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) = {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩})

Distinct variable group:   𝑒,𝑁,𝑥

Proof of Theorem stgrfv

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-stgr 47854	. . 3 ⊢ StarGr = (𝑛 ∈ ℕ0 ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
2	1	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ0 → StarGr = (𝑛 ∈ ℕ0 ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩}))
3		oveq2 7438	. . . . 5 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	opeq2d 4884	. . . 4 ⊢ (𝑛 = 𝑁 → ⟨(Base‘ndx), (0...𝑛)⟩ = ⟨(Base‘ndx), (0...𝑁)⟩)
5	3	pweqd 4621	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝒫 (0...𝑛) = 𝒫 (0...𝑁))
6		oveq2 7438	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
7	6	rexeqdv 3324	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}))
8	5, 7	rabeqbidv 3451	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}} = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
9	8	reseq2d 5999	. . . . 5 ⊢ (𝑛 = 𝑁 → ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}}) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
10	9	opeq2d 4884	. . . 4 ⊢ (𝑛 = 𝑁 → ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩ = ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩)
11	4, 10	preq12d 4745	. . 3 ⊢ (𝑛 = 𝑁 → {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩} = {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩})
12	11	adantl 481	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 = 𝑁) → {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩} = {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩})
13		id 22	. 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
14		prex 5442	. . 3 ⊢ {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩} ∈ V
15	14	a1i 11	. 2 ⊢ (𝑁 ∈ ℕ0 → {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩} ∈ V)
16	2, 12, 13, 15	fvmptd 7022	1 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) = {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩})

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  {crab 3432  Vcvv 3477  𝒫 cpw 4604  {cpr 4632  ⟨cop 4636   ↦ cmpt 5230   I cid 5581   ↾ cres 5690  ‘cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153  ℕ₀cn0 12523  ...cfz 13543  ndxcnx 17226  Basecbs 17244  .efcedgf 29017  StarGrcstgr 47853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-stgr 47854

This theorem is referenced by:  stgrvtx  47856  stgriedg  47857  stgr0  47862  stgr1  47863