Theorem om2uz0i 13983

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 6883	. 2 ⊢ (𝐺‘∅) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅)
3		om2uz.1	. . 3 ⊢ 𝐶 ∈ ℤ
4		fr0g 8423	. . 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 2792	1 ⊢ (𝐺‘∅) = 𝐶

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294   ↦ cmpt 5196   ↾ cres 5664  ‘cfv 6537  (class class class)co 7411  ωcom 7862  reccrdg 8396  1c1 11101   + caddc 11103  ℤcz 12591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397

This theorem is referenced by:  om2uzuzi  13985  om2uzrani  13988  om2uzrdg  13992  uzrdgxfr  14003  fzennn  14004  axdc4uzlem  14019  hashgadd  14413