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Theorem nfrdg 7035
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 7031 . 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 2562 . . . 4  |-  F/_ x _V
3 nfv 1726 . . . . 5  |-  F/ x  g  =  (/)
4 nfrdg.2 . . . . 5  |-  F/_ x A
5 nfv 1726 . . . . . 6  |-  F/ x Lim  dom  g
6 nfcv 2562 . . . . . 6  |-  F/_ x U. ran  g
7 nfrdg.1 . . . . . . 7  |-  F/_ x F
8 nfcv 2562 . . . . . . 7  |-  F/_ x
( g `  U. dom  g )
97, 8nffv 5810 . . . . . 6  |-  F/_ x
( F `  (
g `  U. dom  g
) )
105, 6, 9nfif 3911 . . . . 5  |-  F/_ x if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) )
113, 4, 10nfif 3911 . . . 4  |-  F/_ x if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )
122, 11nfmpt 4480 . . 3  |-  F/_ x
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )
1312nfrecs 6999 . 2  |-  F/_ xrecs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
141, 13nfcxfr 2560 1  |-  F/_ x rec ( F ,  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1403   F/_wnfc 2548   _Vcvv 3056   (/)c0 3735   ifcif 3882   U.cuni 4188    |-> cmpt 4450   Lim wlim 4820   dom cdm 4940   ran crn 4941   ` cfv 5523  recscrecs 6996   reccrdg 7030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-iota 5487  df-fv 5531  df-recs 6997  df-rdg 7031
This theorem is referenced by:  rdgsucmptf  7049  rdgsucmptnf  7050  frsucmpt  7058  frsucmptn  7059  nfseq  12069  trpredlem1  29978  trpredrec  29989
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