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Theorem rdgisuc1 6249
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 6250. (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 4387 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
53, 4syl 14 . . 3  |-  ( ph  ->  suc  B  e.  On )
6 rdgival 6247 . . 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 1201 . 2  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) ) )
8 df-suc 4263 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
9 iuneq1 3796 . . . . . . 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 3862 . . . . . 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 2138 . . . . 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 5389 . . . . . . . 8  |-  ( x  =  B  ->  ( rec ( F ,  A
) `  x )  =  ( rec ( F ,  A ) `  B ) )
1413fveq2d 5393 . . . . . . 7  |-  ( x  =  B  ->  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1514iunxsng 3858 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1615uneq2d 3200 . . . . 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 2162 . . . 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 3200 . . 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 2150 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 1316    e. wcel 1465   _Vcvv 2660    u. cun 3039   {csn 3497   U_ciun 3783   Oncon0 4255   suc csuc 4257    Fn wfn 5088   ` cfv 5093   reccrdg 6234
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170  df-irdg 6235
This theorem is referenced by:  rdgisucinc  6250
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