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Theorem rdgisuc1 6361
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 6362. (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 4483 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  On )
53, 4syl 14 . . 3  |-  ( ph  ->  suc  B  e.  On )
6 rdgival 6359 . . 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 1233 . 2  |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  U_ x  e.  suc  B ( F `
 ( rec ( F ,  A ) `  x ) ) ) )
8 df-suc 4354 . . . . . . 7  |-  suc  B  =  ( B  u.  { B } )
9 iuneq1 3884 . . . . . . 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 3950 . . . . . 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 2191 . . . . 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 5494 . . . . . . . 8  |-  ( x  =  B  ->  ( rec ( F ,  A
) `  x )  =  ( rec ( F ,  A ) `  B ) )
1413fveq2d 5498 . . . . . . 7  |-  ( x  =  B  ->  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1514iunxsng 3946 . . . . . 6  |-  ( B  e.  On  ->  U_ x  e.  { B }  ( F `  ( rec ( F ,  A ) `
 x ) )  =  ( F `  ( rec ( F ,  A ) `  B
) ) )
1615uneq2d 3281 . . . . 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 2215 . . . 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 3281 . . 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 2203 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 1348    e. wcel 2141   _Vcvv 2730    u. cun 3119   {csn 3581   U_ciun 3871   Oncon0 4346   suc csuc 4348    Fn wfn 5191   ` cfv 5196   reccrdg 6346
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 609  ax-in2 610  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-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-recs 6282  df-irdg 6347
This theorem is referenced by:  rdgisucinc  6362
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