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Theorem nfrdg 6664
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 6660 . 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 2571 . . . 4  |-  F/_ x _V
3 nfv 1629 . . . . 5  |-  F/ x  g  =  (/)
4 nfrdg.2 . . . . 5  |-  F/_ x A
5 nfv 1629 . . . . . 6  |-  F/ x Lim  dom  g
6 nfcv 2571 . . . . . 6  |-  F/_ x U. ran  g
7 nfrdg.1 . . . . . . 7  |-  F/_ x F
8 nfcv 2571 . . . . . . 7  |-  F/_ x
( g `  U. dom  g )
97, 8nffv 5727 . . . . . 6  |-  F/_ x
( F `  (
g `  U. dom  g
) )
105, 6, 9nfif 3755 . . . . 5  |-  F/_ x if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) )
113, 4, 10nfif 3755 . . . 4  |-  F/_ x if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )
122, 11nfmpt 4289 . . 3  |-  F/_ x
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )
1312nfrecs 6627 . 2  |-  F/_ xrecs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
141, 13nfcxfr 2568 1  |-  F/_ x rec ( F ,  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   F/_wnfc 2558   _Vcvv 2948   (/)c0 3620   ifcif 3731   U.cuni 4007    e. cmpt 4258   Lim wlim 4574   dom cdm 4870   ran crn 4871   ` cfv 5446  recscrecs 6624   reccrdg 6659
This theorem is referenced by:  rdgsucmptf  6678  rdgsucmptnf  6679  frsucmpt  6687  frsucmptn  6688  nfseq  11325  trpredlem1  25497  trpredrec  25508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-iota 5410  df-fv 5454  df-recs 6625  df-rdg 6660
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