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Theorem nffrec 6013
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 6009 . 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 2194 . . . . 5  |-  F/_ x _V
3 nfcv 2194 . . . . . . . 8  |-  F/_ x om
4 nfv 1437 . . . . . . . . 9  |-  F/ x dom  g  =  suc  m
5 nffrec.1 . . . . . . . . . . 11  |-  F/_ x F
6 nfcv 2194 . . . . . . . . . . 11  |-  F/_ x
( g `  m
)
75, 6nffv 5213 . . . . . . . . . 10  |-  F/_ x
( F `  (
g `  m )
)
87nfcri 2188 . . . . . . . . 9  |-  F/ x  y  e.  ( F `  ( g `  m
) )
94, 8nfan 1473 . . . . . . . 8  |-  F/ x
( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )
103, 9nfrexya 2380 . . . . . . 7  |-  F/ x E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) )
11 nfv 1437 . . . . . . . 8  |-  F/ x dom  g  =  (/)
12 nffrec.2 . . . . . . . . 9  |-  F/_ x A
1312nfcri 2188 . . . . . . . 8  |-  F/ x  y  e.  A
1411, 13nfan 1473 . . . . . . 7  |-  F/ x
( dom  g  =  (/) 
/\  y  e.  A
)
1510, 14nfor 1482 . . . . . 6  |-  F/ x
( E. m  e. 
om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) )
1615nfab 2198 . . . . 5  |-  F/_ x { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) }
172, 16nfmpt 3877 . . . 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 5953 . . 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 4642 . 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 2191 1  |-  F/_ xfrec ( F ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    \/ wo 639    = wceq 1259    e. wcel 1409   {cab 2042   F/_wnfc 2181   E.wrex 2324   _Vcvv 2574   (/)c0 3252    |-> cmpt 3846   suc csuc 4130   omcom 4341   dom cdm 4373    |` cres 4375   ` cfv 4930  recscrecs 5950  freccfrec 6008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-xp 4379  df-res 4385  df-iota 4895  df-fv 4938  df-recs 5951  df-frec 6009
This theorem is referenced by:  nfiseq  9382
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