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Theorem freceq2 6372
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 108 . . . . . . . . 9  |-  ( ( A  =  B  /\  g  e.  _V )  ->  A  =  B )
21eleq2d 2240 . . . . . . . 8  |-  ( ( A  =  B  /\  g  e.  _V )  ->  ( x  e.  A  <->  x  e.  B ) )
32anbi2d 461 . . . . . . 7  |-  ( ( A  =  B  /\  g  e.  _V )  ->  ( ( dom  g  =  (/)  /\  x  e.  A )  <->  ( dom  g  =  (/)  /\  x  e.  B ) ) )
43orbi2d 785 . . . . . 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 2288 . . . . 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 4079 . . . 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 6285 . . . 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 4890 . 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 6370 . 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 6370 . 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 2228 1  |-  ( A  =  B  -> frec ( F ,  A )  = frec ( F ,  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 703    = wceq 1348    e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730   (/)c0 3414    |-> cmpt 4050   suc csuc 4350   omcom 4574   dom cdm 4611    |` cres 4613   ` cfv 5198  recscrecs 6283  freccfrec 6369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-res 4623  df-iota 5160  df-fv 5206  df-recs 6284  df-frec 6370
This theorem is referenced by:  seqeq1  10404  seqeq3  10406
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