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Theorem om2uz0i 11857
Description: The mapping  G is a one-to-one mapping from  om 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  NN0 or 1 for the upper integers  NN), 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  NN0 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  e.  ZZ
om2uz.2  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
Assertion
Ref Expression
om2uz0i  |-  ( G `
 (/) )  =  C
Distinct variable group:    x, C
Allowed substitution hint:    G( x)

Proof of Theorem om2uz0i
StepHypRef Expression
1 om2uz.2 . . 3  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
21fveq1i 5776 . 2  |-  ( G `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )
3 om2uz.1 . . 3  |-  C  e.  ZZ
4 fr0g 6977 . . 3  |-  ( C  e.  ZZ  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C )
53, 4ax-mp 5 . 2  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C
62, 5eqtri 2478 1  |-  ( G `
 (/) )  =  C
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1757   _Vcvv 3054   (/)c0 3721    |-> cmpt 4434    |` cres 4926   ` cfv 5502  (class class class)co 6176   omcom 6562   reccrdg 6951   1c1 9370    + caddc 9372   ZZcz 10733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-om 6563  df-recs 6918  df-rdg 6952
This theorem is referenced by:  om2uzuzi  11859  om2uzrani  11862  om2uzrdg  11866  uzrdgxfr  11876  fzennn  11877  axdc4uzlem  11891  hashgadd  12228
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