Theorem om2uz0i 12786

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 6230	. 2 ⊢ (𝐺‘∅) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅)
3		om2uz.1	. . 3 ⊢ 𝐶 ∈ ℤ
4		fr0g 7576	. . 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 2673	1 ⊢ (𝐺‘∅) = 𝐶

Syntax hints:   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948   ↦ cmpt 4762   ↾ cres 5145  ‘cfv 5926  (class class class)co 6690  ωcom 7107  reccrdg 7550  1c1 9975   + caddc 9977  ℤcz 11415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551

This theorem is referenced by:  om2uzuzi  12788  om2uzrani  12791  om2uzrdg  12795  uzrdgxfr  12806  fzennn  12807  axdc4uzlem  12822  hashgadd  13204