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Theorem nffrec 6115
Description: Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Hypotheses
Ref Expression
nffrec.1  |-  F/_ x F
nffrec.2  |-  F/_ x A
Assertion
Ref Expression
nffrec  |-  F/_ xfrec ( F ,  A )

Proof of Theorem nffrec
Dummy variables  g  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frec 6110 . 2  |- frec ( F ,  A )  =  (recs ( ( g  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } ) )  |`  om )
2 nfcv 2225 . . . . 5  |-  F/_ x _V
3 nfcv 2225 . . . . . . . 8  |-  F/_ x om
4 nfv 1464 . . . . . . . . 9  |-  F/ x dom  g  =  suc  m
5 nffrec.1 . . . . . . . . . . 11  |-  F/_ x F
6 nfcv 2225 . . . . . . . . . . 11  |-  F/_ x
( g `  m
)
75, 6nffv 5278 . . . . . . . . . 10  |-  F/_ x
( F `  (
g `  m )
)
87nfcri 2219 . . . . . . . . 9  |-  F/ x  y  e.  ( F `  ( g `  m
) )
94, 8nfan 1500 . . . . . . . 8  |-  F/ x
( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )
103, 9nfrexya 2413 . . . . . . 7  |-  F/ x E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) )
11 nfv 1464 . . . . . . . 8  |-  F/ x dom  g  =  (/)
12 nffrec.2 . . . . . . . . 9  |-  F/_ x A
1312nfcri 2219 . . . . . . . 8  |-  F/ x  y  e.  A
1411, 13nfan 1500 . . . . . . 7  |-  F/ x
( dom  g  =  (/) 
/\  y  e.  A
)
1510, 14nfor 1509 . . . . . 6  |-  F/ x
( E. m  e. 
om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) )
1615nfab 2229 . . . . 5  |-  F/_ x { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) }
172, 16nfmpt 3905 . . . 4  |-  F/_ x
( g  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
1817nfrecs 6026 . . 3  |-  F/_ xrecs ( ( g  e. 
_V  |->  { y  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } ) )
1918, 3nfres 4683 . 2  |-  F/_ x
(recs ( ( g  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } ) )  |`  om )
201, 19nfcxfr 2222 1  |-  F/_ xfrec ( F ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662    = wceq 1287    e. wcel 1436   {cab 2071   F/_wnfc 2212   E.wrex 2356   _Vcvv 2615   (/)c0 3275    |-> cmpt 3874   suc csuc 4166   omcom 4378   dom cdm 4411    |` cres 4413   ` cfv 4981  recscrecs 6023  freccfrec 6109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-un 2992  df-in 2994  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-xp 4417  df-res 4423  df-iota 4946  df-fv 4989  df-recs 6024  df-frec 6110
This theorem is referenced by:  nfiseq  9786
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