Theorem stgrfv 48580

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 48579	. . 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 7406	. . . . 5 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	opeq2d 4840	. . . 4 ⊢ (𝑛 = 𝑁 → ⟨(Base‘ndx), (0...𝑛)⟩ = ⟨(Base‘ndx), (0...𝑁)⟩)
5	3	pweqd 4574	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝒫 (0...𝑛) = 𝒫 (0...𝑁))
6		oveq2 7406	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
7	6	rexeqdv 3323	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}))
8	5, 7	rabeqbidv 3434	. . . . . 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 5967	. . . . 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 4840	. . . 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 4702	. . 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 485	. 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 5397	. . 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 6985	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 1562   ∈ wcel 2144  ∃wrex 3088  {crab 3416  Vcvv 3456  𝒫 cpw 4557  {cpr 4586  ⟨cop 4590   ↦ cmpt 5183   I cid 5543   ↾ cres 5651  ‘cfv 6523  (class class class)co 7398  0cc0 11075  1c1 11076  ℕ₀cn0 12483  ...cfz 13514  ndxcnx 17231  Basecbs 17247  .efcedgf 29191  StarGrcstgr 48578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-res 5661  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-stgr 48579

This theorem is referenced by:  stgrvtx  48581  stgriedg  48582  stgr0  48587  stgr1  48588