Theorem stgrfv 48445

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 48444	. . 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 7370	. . . . 5 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	opeq2d 4824	. . . 4 ⊢ (𝑛 = 𝑁 → ⟨(Base‘ndx), (0...𝑛)⟩ = ⟨(Base‘ndx), (0...𝑁)⟩)
5	3	pweqd 4559	. . . . . . 7 ⊢ (𝑛 = 𝑁 → 𝒫 (0...𝑛) = 𝒫 (0...𝑁))
6		oveq2 7370	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
7	6	rexeqdv 3297	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥} ↔ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}))
8	5, 7	rabeqbidv 3408	. . . . . 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 5940	. . . . 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 4824	. . . 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 4686	. . 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 5377	. . 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 6951	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 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390  Vcvv 3430  𝒫 cpw 4542  {cpr 4570  ⟨cop 4574   ↦ cmpt 5167   I cid 5520   ↾ cres 5628  ‘cfv 6494  (class class class)co 7362  0cc0 11033  1c1 11034  ℕ₀cn0 12432  ...cfz 13456  ndxcnx 17158  Basecbs 17174  .efcedgf 29075  StarGrcstgr 48443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-res 5638  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-stgr 48444

This theorem is referenced by:  stgrvtx  48446  stgriedg  48447  stgr0  48452  stgr1  48453