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Theorem freceq2 6393
Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
freceq2  |-  ( A  =  B  -> frec ( F ,  A )  = frec ( F ,  B
) )

Proof of Theorem freceq2
Dummy variables  x  g  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 109 . . . . . . . . 9  |-  ( ( A  =  B  /\  g  e.  _V )  ->  A  =  B )
21eleq2d 2247 . . . . . . . 8  |-  ( ( A  =  B  /\  g  e.  _V )  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi2d 464 . . . . . . 7  |-  ( ( A  =  B  /\  g  e.  _V )  ->  ( ( dom  g  =  (/)  /\  x  e.  A )  <->  ( dom  g  =  (/)  /\  x  e.  B ) ) )
43orbi2d 790 . . . . . 6  |-  ( ( A  =  B  /\  g  e.  _V )  ->  ( ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) ) )
54abbidv 2295 . . . . 5  |-  ( ( A  =  B  /\  g  e.  _V )  ->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  =  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } )
65mpteq2dva 4093 . . . 4  |-  ( A  =  B  ->  (
g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } ) )
7 recseq 6306 . . . 4  |-  ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } )  -> recs ( (
g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  = recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } ) ) )
86, 7syl 14 . . 3  |-  ( A  =  B  -> recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  = recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } ) ) )
98reseq1d 4906 . 2  |-  ( A  =  B  ->  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } ) )  |`  om )
)
10 df-frec 6391 . 2  |- frec ( F ,  A )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
11 df-frec 6391 . 2  |- frec ( F ,  B )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  B ) ) } ) )  |`  om )
129, 10, 113eqtr4g 2235 1  |-  ( A  =  B  -> frec ( F ,  A )  = frec ( F ,  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2737   (/)c0 3422    |-> cmpt 4064   suc csuc 4365   omcom 4589   dom cdm 4626    |` cres 4628   ` cfv 5216  recscrecs 6304  freccfrec 6390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-res 4638  df-iota 5178  df-fv 5224  df-recs 6305  df-frec 6391
This theorem is referenced by:  seqeq1  10447  seqeq3  10449
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