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Theorem nfrdg 6601
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 6597 . 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 2516 . . . 4  |-  F/_ x _V
3 nfv 1626 . . . . 5  |-  F/ x  g  =  (/)
4 nfrdg.2 . . . . 5  |-  F/_ x A
5 nfv 1626 . . . . . 6  |-  F/ x Lim  dom  g
6 nfcv 2516 . . . . . 6  |-  F/_ x U. ran  g
7 nfrdg.1 . . . . . . 7  |-  F/_ x F
8 nfcv 2516 . . . . . . 7  |-  F/_ x
( g `  U. dom  g )
97, 8nffv 5668 . . . . . 6  |-  F/_ x
( F `  (
g `  U. dom  g
) )
105, 6, 9nfif 3699 . . . . 5  |-  F/_ x if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) )
113, 4, 10nfif 3699 . . . 4  |-  F/_ x if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )
122, 11nfmpt 4231 . . 3  |-  F/_ x
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )
1312nfrecs 6564 . 2  |-  F/_ xrecs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
141, 13nfcxfr 2513 1  |-  F/_ x rec ( F ,  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649   F/_wnfc 2503   _Vcvv 2892   (/)c0 3564   ifcif 3675   U.cuni 3950    e. cmpt 4200   Lim wlim 4516   dom cdm 4811   ran crn 4812   ` cfv 5387  recscrecs 6561   reccrdg 6596
This theorem is referenced by:  rdgsucmptf  6615  rdgsucmptnf  6616  frsucmpt  6624  frsucmptn  6625  nfseq  11253  trpredlem1  25247  trpredrec  25258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-iota 5351  df-fv 5395  df-recs 6562  df-rdg 6597
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