Theorem stgrfv 48141

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 48140	. . 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 7364	. . . . 5 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	opeq2d 4834	. . . 4 ⊢ (𝑛 = 𝑁 → ⟨(Base‘ndx), (0...𝑛)⟩ = ⟨(Base‘ndx), (0...𝑁)⟩)
5	3	pweqd 4569	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝒫 (0...𝑛) = 𝒫 (0...𝑁))
6		oveq2 7364	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
7	6	rexeqdv 3295	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}))
8	5, 7	rabeqbidv 3415	. . . . . 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 5936	. . . . 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 4834	. . . 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 4696	. . 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 5380	. . 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 6946	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 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397  Vcvv 3438  𝒫 cpw 4552  {cpr 4580  ⟨cop 4584   ↦ cmpt 5177   I cid 5516   ↾ cres 5624  ‘cfv 6490  (class class class)co 7356  0cc0 11024  1c1 11025  ℕ₀cn0 12399  ...cfz 13421  ndxcnx 17118  Basecbs 17134  .efcedgf 29010  StarGrcstgr 48139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-stgr 48140

This theorem is referenced by:  stgrvtx  48142  stgriedg  48143  stgr0  48148  stgr1  48149