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Theorem nfrdg 6427
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 6423 . 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 2419 . . . 4  |-  F/_ x _V
3 nfv 1605 . . . . 5  |-  F/ x  g  =  (/)
4 nfrdg.2 . . . . 5  |-  F/_ x A
5 nfv 1605 . . . . . 6  |-  F/ x Lim  dom  g
6 nfcv 2419 . . . . . 6  |-  F/_ x U. ran  g
7 nfrdg.1 . . . . . . 7  |-  F/_ x F
8 nfcv 2419 . . . . . . 7  |-  F/_ x
( g `  U. dom  g )
97, 8nffv 5532 . . . . . 6  |-  F/_ x
( F `  (
g `  U. dom  g
) )
105, 6, 9nfif 3589 . . . . 5  |-  F/_ x if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) )
113, 4, 10nfif 3589 . . . 4  |-  F/_ x if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )
122, 11nfmpt 4108 . . 3  |-  F/_ x
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )
1312nfrecs 6390 . 2  |-  F/_ xrecs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
141, 13nfcxfr 2416 1  |-  F/_ x rec ( F ,  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   F/_wnfc 2406   _Vcvv 2788   (/)c0 3455   ifcif 3565   U.cuni 3827    e. cmpt 4077   Lim wlim 4393   dom cdm 4689   ran crn 4690   ` cfv 5255  recscrecs 6387   reccrdg 6422
This theorem is referenced by:  rdgsucmptf  6441  rdgsucmptnf  6442  frsucmpt  6450  frsucmptn  6451  axdclem  8146  nfseq  11056  trpredlem1  24230  trpredrec  24241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-recs 6388  df-rdg 6423
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