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Theorem rdgisuc1 6281
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 6282. (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 suceloni 4417 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
53, 4syl 14 . . 3  |-  ( ph  ->  suc  B  e.  On )
6 rdgival 6279 . . 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 1216 . 2  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) ) )
8 df-suc 4293 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
9 iuneq1 3826 . . . . . . 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 3892 . . . . . 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 2160 . . . . 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 5421 . . . . . . . 8  |-  ( x  =  B  ->  ( rec ( F ,  A
) `  x )  =  ( rec ( F ,  A ) `  B ) )
1413fveq2d 5425 . . . . . . 7  |-  ( x  =  B  ->  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1514iunxsng 3888 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1615uneq2d 3230 . . . . 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, 16syl5eq 2184 . . . 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 3230 . . 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 2172 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 1331    e. wcel 1480   _Vcvv 2686    u. cun 3069   {csn 3527   U_ciun 3813   Oncon0 4285   suc csuc 4287    Fn wfn 5118   ` cfv 5123   reccrdg 6266
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202  df-irdg 6267
This theorem is referenced by:  rdgisucinc  6282
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