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Theorem om2uz0i 11006
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 5487 . 2  |-  ( G `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )
3 om2uz.1 . . 3  |-  C  e.  ZZ
4 fr0g 6444 . . 3  |-  ( C  e.  ZZ  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C )
53, 4ax-mp 8 . 2  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C
62, 5eqtri 2304 1  |-  ( G `
 (/) )  =  C
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1685   _Vcvv 2789   (/)c0 3456    e. cmpt 4078   omcom 4655    |` cres 4690   ` cfv 5221  (class class class)co 5820   reccrdg 6418   1c1 8734    + caddc 8736   ZZcz 10020
This theorem is referenced by:  om2uzuzi  11008  om2uzrani  11011  om2uzrdg  11015  uzrdgxfr  11025  fzennn  11026  axdc4uzlem  11040  hashgadd  11355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419
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