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Theorem nfrdg 6443
Description: Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypotheses
Ref Expression
nfrdg.1  |-  F/_ x F
nfrdg.2  |-  F/_ x A
Assertion
Ref Expression
nfrdg  |-  F/_ x rec ( F ,  A
)

Proof of Theorem nfrdg
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 df-rdg 6439 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
2 nfcv 2432 . . . 4  |-  F/_ x _V
3 nfv 1609 . . . . 5  |-  F/ x  g  =  (/)
4 nfrdg.2 . . . . 5  |-  F/_ x A
5 nfv 1609 . . . . . 6  |-  F/ x Lim  dom  g
6 nfcv 2432 . . . . . 6  |-  F/_ x U. ran  g
7 nfrdg.1 . . . . . . 7  |-  F/_ x F
8 nfcv 2432 . . . . . . 7  |-  F/_ x
( g `  U. dom  g )
97, 8nffv 5548 . . . . . 6  |-  F/_ x
( F `  (
g `  U. dom  g
) )
105, 6, 9nfif 3602 . . . . 5  |-  F/_ x if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) )
113, 4, 10nfif 3602 . . . 4  |-  F/_ x if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )
122, 11nfmpt 4124 . . 3  |-  F/_ x
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )
1312nfrecs 6406 . 2  |-  F/_ xrecs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
141, 13nfcxfr 2429 1  |-  F/_ x rec ( F ,  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   F/_wnfc 2419   _Vcvv 2801   (/)c0 3468   ifcif 3578   U.cuni 3843    e. cmpt 4093   Lim wlim 4409   dom cdm 4705   ran crn 4706   ` cfv 5271  recscrecs 6403   reccrdg 6438
This theorem is referenced by:  rdgsucmptf  6457  rdgsucmptnf  6458  frsucmpt  6466  frsucmptn  6467  axdclem  8162  nfseq  11072  trpredlem1  24301  trpredrec  24312
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-iota 5235  df-fv 5279  df-recs 6404  df-rdg 6439
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