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Theorem rdgsucg 7081
Description: The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdgsucg  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )

Proof of Theorem rdgsucg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgdmlim 7075 . . 3  |-  Lim  dom  rec ( F ,  A
)
2 limsuc 6657 . . 3  |-  ( Lim 
dom  rec ( F ,  A )  ->  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) ) )
31, 2ax-mp 5 . 2  |-  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) )
4 eqid 2462 . . 3  |-  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `  U. dom  x ) ) ) ) )  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `
 U. dom  x
) ) ) ) )
5 rdgvalg 7077 . . 3  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  y )  =  ( ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( F `  ( x `
 U. dom  x
) ) ) ) ) `  ( rec ( F ,  A
)  |`  y ) ) )
6 rdgseg 7080 . . 3  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  y )  e.  _V )
7 rdgfun 7074 . . . 4  |-  Fun  rec ( F ,  A )
8 funfn 5610 . . . 4  |-  ( Fun 
rec ( F ,  A )  <->  rec ( F ,  A )  Fn  dom  rec ( F ,  A ) )
97, 8mpbi 208 . . 3  |-  rec ( F ,  A )  Fn  dom  rec ( F ,  A )
10 limord 4932 . . . 4  |-  ( Lim 
dom  rec ( F ,  A )  ->  Ord  dom 
rec ( F ,  A ) )
111, 10ax-mp 5 . . 3  |-  Ord  dom  rec ( F ,  A
)
124, 5, 6, 9, 11tz7.44-2 7065 . 2  |-  ( suc 
B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
133, 12sylbi 195 1  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   _Vcvv 3108   (/)c0 3780   ifcif 3934   U.cuni 4240    |-> cmpt 4500   Ord word 4872   Lim wlim 4874   suc csuc 4875   dom cdm 4994   ran crn 4995   Fun wfun 5575    Fn wfn 5576   ` cfv 5581   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-recs 7034  df-rdg 7068
This theorem is referenced by:  rdgsuc  7082  rdgsucmptnf  7087  frsuc  7094  r1sucg  8178
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