Theorem om2uz0i 13314

Description: The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ₀ or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (This series of theorems generalizes an earlier series for ℕ₀ contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Hypotheses

Ref	Expression
om2uz.1	⊢ 𝐶 ∈ ℤ
om2uz.2	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)

Assertion

Ref	Expression
om2uz0i	⊢ (𝐺‘∅) = 𝐶

Distinct variable group:   𝑥,𝐶

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem om2uz0i

Step	Hyp	Ref	Expression
1		om2uz.2	. . 3 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
2	1	fveq1i 6670	. 2 ⊢ (𝐺‘∅) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅)
3		om2uz.1	. . 3 ⊢ 𝐶 ∈ ℤ
4		fr0g 8070	. . 3 ⊢ (𝐶 ∈ ℤ → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅) = 𝐶)
5	3, 4	ax-mp 5	. 2 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅) = 𝐶
6	2, 5	eqtri 2844	1 ⊢ (𝐺‘∅) = 𝐶

Syntax hints:   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290   ↦ cmpt 5145   ↾ cres 5556  ‘cfv 6354  (class class class)co 7155  ωcom 7579  reccrdg 8044  1c1 10537   + caddc 10539  ℤcz 11980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045

This theorem is referenced by:  om2uzuzi  13316  om2uzrani  13319  om2uzrdg  13323  uzrdgxfr  13334  fzennn  13335  axdc4uzlem  13350  hashgadd  13737