Theorem om2uz0i 13872

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 6835	. 2 ⊢ (𝐺‘∅) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)‘∅)
3		om2uz.1	. . 3 ⊢ 𝐶 ∈ ℤ
4		fr0g 8367	. . 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 2759	1 ⊢ (𝐺‘∅) = 𝐶

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285   ↦ cmpt 5179   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358  ωcom 7808  reccrdg 8340  1c1 11029   + caddc 11031  ℤcz 12490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  om2uzuzi  13874  om2uzrani  13877  om2uzrdg  13881  uzrdgxfr  13892  fzennn  13893  axdc4uzlem  13908  hashgadd  14302