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Theorem om2uz0i 10977
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 5459 . 2  |-  ( G `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )
3 om2uz.1 . . 3  |-  C  e.  ZZ
4 fr0g 6416 . . 3  |-  ( C  e.  ZZ  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C )
53, 4ax-mp 10 . 2  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C
62, 5eqtri 2278 1  |-  ( G `
 (/) )  =  C
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2763   (/)c0 3430    e. cmpt 4051   omcom 4628    |` cres 4663   ` cfv 4673  (class class class)co 5792   reccrdg 6390   1c1 8706    + caddc 8708   ZZcz 9992
This theorem is referenced by:  om2uzuzi  10979  om2uzrani  10982  om2uzrdg  10986  uzrdgxfr  10996  fzennn  10997  axdc4uzlem  11011  hashgadd  11326
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391
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