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Theorem rdglimg 6454
Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdglimg  |-  ( ( B  e.  dom  rec ( F ,  A )  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. ( rec ( F ,  A ) " B
) )

Proof of Theorem rdglimg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . 2  |-  ( 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
) ) ) ) )
2 rdgvalg 6448 . 2  |-  ( 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 ) ) )
3 rdgseg 6451 . 2  |-  ( y  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  y )  e.  _V )
4 rdgfun 6445 . . 3  |-  Fun  rec ( F ,  A )
5 funfn 5299 . . 3  |-  ( Fun 
rec ( F ,  A )  <->  rec ( F ,  A )  Fn  dom  rec ( F ,  A ) )
64, 5mpbi 199 . 2  |-  rec ( F ,  A )  Fn  dom  rec ( F ,  A )
7 rdgdmlim 6446 . . 3  |-  Lim  dom  rec ( F ,  A
)
8 limord 4467 . . 3  |-  ( Lim 
dom  rec ( F ,  A )  ->  Ord  dom 
rec ( F ,  A ) )
97, 8ax-mp 8 . 2  |-  Ord  dom  rec ( F ,  A
)
101, 2, 3, 6, 9tz7.44-3 6437 1  |-  ( ( B  e.  dom  rec ( F ,  A )  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. ( rec ( F ,  A ) " B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ifcif 3578   U.cuni 3843    e. cmpt 4093   Ord word 4407   Lim wlim 4409   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  rdglim  6455  r1limg  7459
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-reu 2563  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-pw 3640  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-lim 4413  df-suc 4414  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-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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