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Theorem rdgsucg 6431
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 )
) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem rdgsucg
StepHypRef Expression
1 rdgdmlim 6425 . . 3  |-  Lim  dom  rec ( F ,  A
)
2 limsuc 4639 . . 3  |-  ( Lim 
dom  rec ( F ,  A )  ->  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) ) )
31, 2ax-mp 10 . 2  |-  ( B  e.  dom  rec ( F ,  A )  <->  suc 
B  e.  dom  rec ( F ,  A ) )
4 eqid 2284 . . 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 6427 . . 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 6430 . . 3  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  y )  e.  _V )
7 rdgfun 6424 . . . 4  |-  Fun  rec ( F ,  A )
8 funfn 5249 . . . 4  |-  ( Fun 
rec ( F ,  A )  <->  rec ( F ,  A )  Fn  dom  rec ( F ,  A ) )
97, 8mpbi 201 . . 3  |-  rec ( F ,  A )  Fn  dom  rec ( F ,  A )
10 limord 4450 . . . 4  |-  ( Lim 
dom  rec ( F ,  A )  ->  Ord  dom 
rec ( F ,  A ) )
111, 10ax-mp 10 . . 3  |-  Ord  dom  rec ( F ,  A
)
124, 5, 6, 9, 11tz7.44-2 6415 . 2  |-  ( suc 
B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
133, 12sylbi 189 1  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685   _Vcvv 2789   (/)c0 3456   ifcif 3566   U.cuni 3828    e. cmpt 4078   Ord word 4390   Lim wlim 4392   suc csuc 4393   dom cdm 4688   ran crn 4689   Fun wfun 5215    Fn wfn 5216   ` cfv 5221   reccrdg 6417
This theorem is referenced by:  rdgsuc  6432  rdgsucmptnf  6437  frsuc  6444  r1sucg  7436
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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-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 6383  df-rdg 6418
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