Theorem om2uz0i 13854

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 6823	. 2 ⊢ (𝐺‘∅) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅)
3		om2uz.1	. . 3 ⊢ 𝐶 ∈ ℤ
4		fr0g 8358	. . 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 2752	1 ⊢ (𝐺‘∅) = 𝐶

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284   ↦ cmpt 5173   ↾ cres 5621  ‘cfv 6482  (class class class)co 7349  ωcom 7799  reccrdg 8331  1c1 11010   + caddc 11012  ℤcz 12471

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  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  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-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332

This theorem is referenced by:  om2uzuzi  13856  om2uzrani  13859  om2uzrdg  13863  uzrdgxfr  13874  fzennn  13875  axdc4uzlem  13890  hashgadd  14284