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Theorem rdgseg 6451
Description: The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
rdgseg  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  B )  e.  _V )

Proof of Theorem rdgseg
Dummy variables  x  y  f  g  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rdg 6439 . . 3  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
21reseq1i 4967 . 2  |-  ( rec ( F ,  A
)  |`  B )  =  (recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) )  |`  B )
3 rdglem1 6444 . . . 4  |-  { w  |  E. y  e.  On  ( w  Fn  y  /\  A. v  e.  y  ( w `  v
)  =  ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  ( w  |`  v ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) `  ( f  |`  y ) ) ) }
43tfrlem9a 6418 . . 3  |-  ( B  e.  dom recs ( (
g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )  ->  (recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )  |`  B )  e.  _V )
51dmeqi 4896 . . 3  |-  dom  rec ( F ,  A )  =  dom recs ( (
g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )
64, 5eleq2s 2388 . 2  |-  ( B  e.  dom  rec ( F ,  A )  ->  (recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) )  |`  B )  e.  _V )
72, 6syl5eqel 2380 1  |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801   (/)c0 3468   ifcif 3578   U.cuni 3843    e. cmpt 4093   Oncon0 4408   Lim wlim 4409   dom cdm 4705   ran crn 4706    |` cres 4707    Fn wfn 5266   ` cfv 5271  recscrecs 6403   reccrdg 6438
This theorem is referenced by:  rdgsucg  6452  rdglimg  6454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-recs 6404  df-rdg 6439
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