Theorem stgrfv 47947

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 47946	. . 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 7357	. . . . 5 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	opeq2d 4831	. . . 4 ⊢ (𝑛 = 𝑁 → ⟨(Base‘ndx), (0...𝑛)⟩ = ⟨(Base‘ndx), (0...𝑁)⟩)
5	3	pweqd 4568	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝒫 (0...𝑛) = 𝒫 (0...𝑁))
6		oveq2 7357	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
7	6	rexeqdv 3290	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}))
8	5, 7	rabeqbidv 3413	. . . . . 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 5930	. . . . 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 4831	. . . 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 4693	. . 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 5376	. . 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 6937	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 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3394  Vcvv 3436  𝒫 cpw 4551  {cpr 4579  ⟨cop 4583   ↦ cmpt 5173   I cid 5513   ↾ cres 5621  ‘cfv 6482  (class class class)co 7349  0cc0 11009  1c1 11010  ℕ₀cn0 12384  ...cfz 13410  ndxcnx 17104  Basecbs 17120  .efcedgf 28933  StarGrcstgr 47945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-stgr 47946

This theorem is referenced by:  stgrvtx  47948  stgriedg  47949  stgr0  47954  stgr1  47955