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Theorem om2uzsuci 11011
Description: The value of  G (see om2uz0i 11010) at a successor. (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
om2uzsuci  |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `
 A )  +  1 ) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    G( x)

Proof of Theorem om2uzsuci
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suceq 4457 . . . 4  |-  ( z  =  A  ->  suc  z  =  suc  A )
21fveq2d 5529 . . 3  |-  ( z  =  A  ->  ( G `  suc  z )  =  ( G `  suc  A ) )
3 fveq2 5525 . . . 4  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
43oveq1d 5873 . . 3  |-  ( z  =  A  ->  (
( G `  z
)  +  1 )  =  ( ( G `
 A )  +  1 ) )
52, 4eqeq12d 2297 . 2  |-  ( z  =  A  ->  (
( G `  suc  z )  =  ( ( G `  z
)  +  1 )  <-> 
( G `  suc  A )  =  ( ( G `  A )  +  1 ) ) )
6 ovex 5883 . . 3  |-  ( ( G `  z )  +  1 )  e. 
_V
7 om2uz.2 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
8 oveq1 5865 . . . 4  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
9 oveq1 5865 . . . 4  |-  ( y  =  ( G `  z )  ->  (
y  +  1 )  =  ( ( G `
 z )  +  1 ) )
107, 8, 9frsucmpt2 6452 . . 3  |-  ( ( z  e.  om  /\  ( ( G `  z )  +  1 )  e.  _V )  ->  ( G `  suc  z )  =  ( ( G `  z
)  +  1 ) )
116, 10mpan2 652 . 2  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
125, 11vtoclga 2849 1  |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `
 A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   suc csuc 4394   omcom 4656    |` cres 4691   ` cfv 5255  (class class class)co 5858   reccrdg 6422   1c1 8738    + caddc 8740   ZZcz 10024
This theorem is referenced by:  om2uzuzi  11012  om2uzlti  11013  om2uzrani  11015  om2uzrdg  11019  uzrdgsuci  11023  uzrdgxfr  11029  fzennn  11030  axdc4uzlem  11044  hashgadd  11359
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423
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