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Theorem rdgisuc1 6409
Description: One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function  F other than  F  Fn  _V. Given that, the resulting expression encompasses both the expected successor term  ( F `  ( rec ( F ,  A ) `  B
) ) but also terms that correspond to the initial value  A and to limit ordinals  U_ x  e.  B ( F `  ( rec ( F ,  A ) `  x
) ).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6410. (Contributed by Jim Kingdon, 9-Jun-2019.)

Hypotheses
Ref Expression
rdgisuc1.1  |-  ( ph  ->  F  Fn  _V )
rdgisuc1.2  |-  ( ph  ->  A  e.  V )
rdgisuc1.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
rdgisuc1  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
Distinct variable groups:    x, F    x, A    x, B    x, V
Allowed substitution hint:    ph( x)

Proof of Theorem rdgisuc1
StepHypRef Expression
1 rdgisuc1.1 . . 3  |-  ( ph  ->  F  Fn  _V )
2 rdgisuc1.2 . . 3  |-  ( ph  ->  A  e.  V )
3 rdgisuc1.3 . . . 4  |-  ( ph  ->  B  e.  On )
4 onsuc 4518 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
53, 4syl 14 . . 3  |-  ( ph  ->  suc  B  e.  On )
6 rdgival 6407 . . 3  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  suc  B  e.  On )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `  ( rec ( F ,  A
) `  x )
) ) )
71, 2, 5, 6syl3anc 1249 . 2  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) ) )
8 df-suc 4389 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
9 iuneq1 3914 . . . . . . 7  |-  ( suc 
B  =  ( B  u.  { B }
)  ->  U_ x  e. 
suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  = 
U_ x  e.  ( B  u.  { B } ) ( F `
 ( rec ( F ,  A ) `  x ) ) )
108, 9ax-mp 5 . . . . . 6  |-  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  = 
U_ x  e.  ( B  u.  { B } ) ( F `
 ( rec ( F ,  A ) `  x ) )
11 iunxun 3981 . . . . . 6  |-  U_ x  e.  ( B  u.  { B } ) ( F `
 ( rec ( F ,  A ) `  x ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x
) )  u.  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `  x
) ) )
1210, 11eqtri 2210 . . . . 5  |-  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x
) )  u.  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `  x
) ) )
13 fveq2 5534 . . . . . . . 8  |-  ( x  =  B  ->  ( rec ( F ,  A
) `  x )  =  ( rec ( F ,  A ) `  B ) )
1413fveq2d 5538 . . . . . . 7  |-  ( x  =  B  ->  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1514iunxsng 3977 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1615uneq2d 3304 . . . . 5  |-  ( B  e.  On  ->  ( U_ x  e.  B  ( F `  ( rec ( F ,  A
) `  x )
)  u.  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) )
1712, 16eqtrid 2234 . . . 4  |-  ( B  e.  On  ->  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) )  =  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x
) )  u.  ( F `  ( rec ( F ,  A ) `
 B ) ) ) )
1817uneq2d 3304 . . 3  |-  ( B  e.  On  ->  ( A  u.  U_ x  e. 
suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A
) `  x )
)  u.  ( F `
 ( rec ( F ,  A ) `  B ) ) ) ) )
193, 18syl 14 . 2  |-  ( ph  ->  ( A  u.  U_ x  e.  suc  B ( F `  ( rec ( F ,  A
) `  x )
) )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
207, 19eqtrd 2222 1  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `
 x ) )  u.  ( F `  ( rec ( F ,  A ) `  B
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   _Vcvv 2752    u. cun 3142   {csn 3607   U_ciun 3901   Oncon0 4381   suc csuc 4383    Fn wfn 5230   ` cfv 5235   reccrdg 6394
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-recs 6330  df-irdg 6395
This theorem is referenced by:  rdgisucinc  6410
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